TEORETYCZNE PODSTAWY JAKO FUNDAMENT

Transkrypt

POLSKO-AMERYKAŃSKIE SYMPOZJUM
Z OKAZJI 75. URODZIN PROFESORA
MIŁOSZA PIOTRA WNUKA ORAZ 50. ROCZNICY
JEGO PRACY NAUKOWEJ I DYDAKTYCZNEJ
TEORETYCZNE PODSTAWY JAKO
FUNDAMENT DOSKONAŁOŚCI
WSPÓŁCZESNYCH TECHNOLOGII
12 września 2011 r.
INSTYTUT ODLEWNICTWA
KRAKÓW 2011
Redaktor Naczelny: Jerzy J. Sobczak
Zespół redakcyjny: Jerzy J. Sobczak, Joanna Madej,
Marta Konieczna
Skład komputerowy: Anna Samek-Bugno
Projekt okładki: Jan Witkowski
© Copyright Instytut Odlewnictwa – Kraków 2011
All rights reserved
ISBN 978-83-88770-65-4
Wydawnictwo:
Instytut Odlewnictwa
ul. Zakopiańska 73
30-418 Kraków
www.iod.krakow.pl
Druk i oprawa:
Instytut Odlewnictwa
POLISH-AMERICAN SYMPOSIUM HONORING
75TH BIRTHDAY AND FIFTY YEARS OF TEACHING
AND RESEARCH OF PROFESSOR MICHAEL P. WNUK
THEORETICAL FOUNDATION OF
EXCELLENCE IN CONTEMPORARY
ENGINEERING
12 September 2011
FOUNDRY RESEARCH INSTITUTE
KRAKOW 2011
Editor: Jerzy J. Sobczak
Editorial staff: Jerzy J. Sobczak, Joanna Madej,
Marta Konieczna
Computer typesetting: Anna Samek-Bugno
Cover design: Jan Witkowski
© Copyright by Foundry Research Institute – Krakow 2011
All rights reserved
ISBN 978-83-88770-65-4
Edited by:
Foundry Research Institute
73 Zakopianska Str.
30-418 Krakow, Poland
www.iod.krakow.pl
Printing:
Foundry Research Institute
Spis treści/Content
1.
Komitet Naukowy/Komitet Organizacyjny ……………………………….
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2.
Scientific Committee/Organizing Committee ………………………….
8
3.
Program Sympozjum ..………………………………………………………………
9
4.
Program of Symposium ……………………………………………………………
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5.
Słowo wstępne …………………………………………………………………………
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6.
Foreword ……………………………………………………………………………………
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7.
Profesor dr hab. Miłosz Piotr Wnuk – Curriculum Vitae …………
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8.
Dr. Michael P. Wnuk – Biographical Information ………………….…
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9.
Michael P. Wnuk – Anegdotycznie o Wnuku ……………………………
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10.
Michael P. Wnuk – 75 years of life and 50 years of teaching
and research …………………………………………………………………………….
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11.
Michael P. Wnuk – Sample Technical Publication .………………….
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12.
Michael P. Wnuk – Kwantowa teoria propagacji
quasi-statycznych szczelin w ośrodkach niesprężystych ……….
31
13.
Michael P. Wnuk – Mathematics of Cassini’s Journey to Saturn
(1997–2004) ……………………………………………………………………………
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14.
Michael P. Wnuk – Mechanics of time dependent fracture ……
73
15.
Letters ……………………………………………………………………………………….
Prof. dr hab. Miłosz Piotr Wnuk
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75 lat i 50 lat pracy badawczej
Komitet Naukowy
(alfabetycznie)
1. Prof. dr hab. inż. Artur Ganczarski – Wydział Mechaniczny,
Politechnika Krakowska, Kraków
2. Prof. dr hab. Jerzy Z. Hubert – Instytut Fizyki Jądrowej, Polska
Akademia Nauk, Kraków
3. Prof. dr hab. inż. Jerzy Pacyna – Katedra Metaloznawstwa
i Metalurgii Proszków, Akademia Górniczo-Hutnicza, Kraków
4. Prof. dr hab. inż. Stanisław Pytko – Wydział Inżynierii Mechanicznej
i Robotyki, Akademia Górniczo-Hutnicza, Kraków
5. Prof. dr hab. inż. Jerzy
Odlewnictwa, Kraków
J.
Sobczak
–
Dyrektor
Instytutu
1. Prof. dr hab. inż. Jerzy J. Sobczak
Odlewnictwa, Kraków – Przewodniczący
–
Dyrektor
Instytutu
Komitet Organizacyjny
(alfabetycznie)
2. Inż. Marta Konieczna – Centrum Informacji i Promocji, Kraków
3. Mgr Joanna Madej – Centrum Informacji i Promocji, Kraków
4. Mgr Anna Samek-Bugno – Centrum Informacji i Promocji, Kraków
5. Mgr inż. Jan Witkowski – Centrum Badań Wysokotemperaturowych,
Kraków
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Scientific Committee
(alphabetically)
1. Prof. Artur Ganczarski, DSc. PhD. Eng. – Department of Mechanical
Engineering, Cracow University of Technology, Krakow
2. Prof. Jerzy Z. Hubert, DSc. PhD. – Department of Structural Research,
Institute of Nuclear Physics, Polish Academy of Sciences, Krakow
3. Prof. Jerzy Pacyna, DSc. PhD. Eng. – Institute of Metallographic and
Materials Engineering, AGH – University of Science and Technology,
Krakow
4. Prof. Stanisław Pytko, DSc. PhD. Eng. – Professor at the Faculty of
Mechanical Engineering and Robotics, AGH – University of Science and
Technology, Krakow
5. Prof. Jerzy J. Sobczak, DSc. PhD. Eng. – Director of the Foundry
Research Institute, Krakow
Organizing Committee
(alphabetically)
1. Prof. Jerzy J. Sobczak, DSc. PhD. Eng. – Director of the Foundry
Research Institute, Krakow – Chair
2. Marta Konieczna, Eng. – Centre of Information and Promotion,
Krakow
3. Joanna Madej, MA. – Centre of Information and Promotion, Krakow
4. Anna Samek-Bugno, MA. – Centre of Information and Promotion,
Krakow
5. Jan Witkowski, MSc. Eng., MA. – Centre for High-temperature
Studies, Krakow
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Program Sympozjum
1. „Ewolucja pól mikro-uszkodzeń w materiałach napromieniowanych
i poddanych plastycznym odkształceniom” – Błażej Skoczeń, Dyrektor Instytutu Mechaniki Stosowanej, Politechnika Krakowska, Kraków, Polska.
2. „Modelowanie powierzchni granicznych dla poprzecznie izotropowych kompozytów SCS-6/Ti-15-3” – Artur Ganczarski, Profesor
Wydziału Mechanicznego Politechniki Krakowskiej, Kraków, Polska.
3. „Procesy anty-entropijne drogą do doskonałości w przyrodzie a także w życiu i pracy Miłosza Piotra Wnuka” – Jerzy Z. Hubert, Profesor nadzwyczajny, Wydział Badań Strukturalnych, Instytut Fizyki
Jądrowej, Polska Akademia Nauk, Kraków, Polska.
4. „Granice stosowalności termodynamiki w zastosowaniach w mechanice” – Arthur Shavit, Profesor Wydziału Mechanicznego Izraelskiego Instytutu Technologii – Technion, Haifa, Izrael.
5. „Przegląd współczesnych technologii związanych ze zmęczeniem
metali” – Stanisław Pytko, Profesor Instytutu Inżynierii Mechanicznej i Robotyki, AGH – Akademia Górniczo-Hutnicza, Kraków, Polska.
6. „Rozwój metod eksperymentalnych używanych w mechanice pękania” – Jerzy Pacyna, Profesor Metaloznawstwa, AGH – Akademia
Górniczo-Hutnicza, Kraków, Polska.
7. „Pod- i ponaddźwiękowa propagacja szczelin w ciałach stałych”,
Ares Rosakis – Dziekan Wydziału Mechanicznego i Aeronautyki, Kalifornijski Instytut Technologiczny – Caltech, Pasadena, Kalifornia,
USA.
8. „Zastosowania kryterium Wnuka do badań podkrytycznej propagacji szczelin” – Wiktor M. Pestrikow, Dyrektor Instytutu Informatyki
Petersburskiego Państwowego Uniwersytetu Ekonomii, Sankt Petersburg, Rosja.
9. „Nieliniowa mechanika pękania i jej rozwój w XX i XXI wieku”, Jurij
G. Matvienko – Dyrektor Oddziału Wytrzymałości, Odporności
i Bezpieczeństwa, Instytut Nauki Inżynierii Mechanicznej, Rosyjska
Akademia Nauk, Moskwa, Rosja.
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10. „Współczesne technologie odlewnictwa. Prognozy oraz trendy rozwoju przemysłu odlewniczego w Polsce” – Jerzy J. Sobczak, Dyrektor Instytutu Odlewnictwa, Kraków, Polska.
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Program of Symposium
1. “Evolution of micro-damage fields in the irradiated materials subjected to plastic straining” – Błażej Skoczen, Head of the Institute
of Applied Mechanics, Cracow University of Technology, Krakow,
Poland.
2. “Modeling of limit surfaces for transversely isotropic SCS-6/Ti-15-3”
– Artur Ganczarski, Professor at Mechanical Engineering Department, Cracow University of Technology, Krakow, Poland.
3. “Physical interpretation of fracture process” – Włodzimierz Wójcik,
Head of the Institute of Physics, Mathematics and Applied Informatics, Cracow University of Technology, Krakow, Poland.
4. “Limits of Thermodynamics in applications to Mechanics” – Arthur
Shavit, Professor at the Department of Mechanical Engineering,
Technion – Israel Institute of Technology, Haifa, Israel.
5. “Contemporary view of technologies related to metal fatigue” –
Stanisław Pytko, Professor at the Faculty of Mechanical Engineering
and Robotics, AGH – University of Science and Technology, Krakow,
Poland.
6. “Recent experimental techniques in fracture mechanics” – Jerzy Pacyna, Department of Metals Engineering and Industrial Informatics,
AGH – University of Science and Technology, Krakow, Poland.
7. “Sub- and Supersonic propagation of cracks” – Ares Rosakis, Dean
of Engineering, California Institute of Technology, Pasadena, California, USA.
8. “Applications of Wnuk’s criterion for subcritical crack propagation in
the Mechanics of Fracture” – Viktor M. Pestrikov, Head of Informatics Department of The Saint Petersburg State University of Service
and Economics, St. Petersburg, Russia.
9. “Nonlinear Fracture Mechanics and its applications in 21st century
engineering” – Yury G. Matvienko, Head of Department of Strength,
Survivability and Safety, Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, Russia.
10. “Modern metal casting technologies. Forecast and development
trends in national foundry industry” – Jerzy J. Sobczak, Head of the
Foundry Research Institute, Krakow, Poland.
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Słowo wstępne
Serdecznie witam wszystkich uczestników Polsko-Amerykańskiego Sympozjum,
zorganizowanego z okazji 75. urodzin naszego rodaka, prof. dr hab. inż. Miłosza
Piotra Wnuka, prominentnego uczonego o międzynarodowej reputacji – według
słów profesora Zenona Mroza z Polskiej Akademii Nauk w Warszawie –
„…specjalisty w skali światowej wnoszącego znaczący oryginalny wkład w swojej
dziedzinie badań…”. Profesor, któremu mija właśnie również 50 lat pracy dydaktycznej, w świecie nauki znany jest jako amerykański uczony Michael Wnuk.
Możliwość goszczenia Państwa w murach naszego Instytutu jest dla mnie przywilejem i zaszczytem. Nieczęsto zdarzają się okazje i wydarzenia tego kalibru jak dzisiejsze Polsko-Amerykańskie Sympozjum, któremu nadaliśmy tytuł: „Teoretyczne
podstawy jako fundament doskonałości współczesnych technologii”.
Częstokroć my, inżynierowie-praktycy z pewnym pobłażaniem spoglądamy na teoretyków zajmujących się mechaniką teoretyczną i stosowaną. Każdy jednak przyzna, że bez solidnych podstaw teoretycznych i bez wkładu uczonych w dziedzinie
mechaniki, nie istniałyby dzisiejsze zawansowane materiały i technologie. Poczynając od mistrza Leonardo da Vinci, wielkiego Galileo Galilei i sir Isaaca Newtona, to
właśnie mechanika, bez wątpienia Pierwsza Dama Dworu, wspólnie z fizyką wiernie
służąc Królowej wszystkich nauk ścisłych – Matematyce, pozwoliła na tak spektakularny rozwój sztuki inżynierskiej oraz towarzyszących jej nowoczesnych sposobów projektowania i wytwarzania. Bez matematycznych podstaw kształtowanych
poprzez ubiegłe stulecia, od Johannesa Keplera, Gottfrieda Wilhelma Leibniza,
wspomnianego uprzednio Isaaca Newtona i Leonarda Eulera do Alberta Einsteina,
nie mielibyśmy dziś tak wyszukanych zaawansowanych technik i technologii, które
zapewniają współczesnemu człowiekowi dostatnią egzystencję oraz godziwy standard życia.
Profesor Miłosz Wnuk uchodząc w wielu kręgach za postać kontrowersyjną, acz
interesującą i barwną, bez wątpienia rozpoznawany jest na świecie jako jeden
z twórców mechaniki pękania i mechaniki uszkodzeń. Wielokrotnie tę właśnie dziedzinę nauki profesor Wnuk przedstawiał w kraju w trakcie swoich wykładów, zarówno w Instytucie Podstawowych Problemów Techniki w Warszawie, jak i na Akademii Górniczo-Hutniczej w Krakowie. Instytut Odlewnictwa również gościł profesora Wnuka, czy to jako konsultanta czy też wykładowcę dla naszych specjalistów
z zakresu kruchego pękania.
Wybrane informacje, dotyczące wkładu profesora Wnuka w rozwój prezentowanej
przez Niego dziedziny wiedzy zostały zebrane i przedstawione w danej monografii.
Na pierwszych jej stronach znajdą Państwo skrócony życiorys profesora i zwięzły
opis jego działalności dydaktycznej oraz naukowej nie tylko w Stanach Zjednoczonych oraz w Polsce, lecz również w takich krajach jak Wielka Brytania, Niemcy,
Rosja, była Jugosławia, Włochy, Chiny oraz Ukraina. Profesor Michael Wnuk od kilku lat pracuje również jako konsultant Narodowej Agencji Aeronautyki i Przestrzeni
Kosmicznej (NASA). Jego raport, sponsorowany przez NASA na temat misji Cassini’ego na Saturna został również włączony do obecnego wydania monograficznego.
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Profesor Wnuk jest autorem blisko stu publikacji naukowo-technicznych oraz kilku
podręczników dla inżynierów oraz pracowników naukowych, zajmujących się mechaniką pękania. Dwa wydania jego uczelnianego skryptu wydanego na Akademii
Górniczo-Hutniczej, a potem monografii „Podstawy mechaniki pękania” znane są
nie tylko w Polsce ale również w Stanach Zjednoczonych, Wielkiej Brytanii i Rosji.
Profesor jest dożywotnim członkiem Nowojorskiej Akademii Nauk oraz członkiemkorespondentem Brytyjskiego Towarzystwa Filozofii Naturalnej, związanego z Uniwersytetem w Cambridge w Anglii, gdzie w roku 1970 pracował na tym właśnie
uniwersytecie jako „distinguished visiting scholar”. Wypada wspomnieć w tym
miejscu, że pracownikiem Cambridge University był kiedyś nie kto inny jak właśnie
sir Isaac Newton.
Wyniki prac teoretycznych profesora M. Wnuka zyskały w literaturze specjalistycznej uznanie poprzez wprowadzenie „równań” czy też „kryteriów”, noszących Jego
imię. W prezentowanym Państwu niniejszym wydaniu można znaleźć odnośniki do
„równania Wnuka-Rice’a-Sorensena”, „równania Wnuka-Knaussa” czy też do „kryterium Wnuka”.
Na koniec pragnę wyrazić serdeczne podziękowanie panu profesorowi Miłoszowi
Wnukowi za aktywny udział w organizacji i przeprowadzeniu Sympozjum, a przede
wszystkim dojazd do Krakowa z odległego miasta Milwaukee w stanie Wisconsin.
Pomimo ponad czterdziestoletniego pobytu w Stanach Profesor Wnuk wielokrotnie
podkreśla, że czuje się bardziej Polakiem niż Amerykaninem. Niekiedy wydaje się
nawet, że profesor jest bardziej propolski, a już na pewno bardziej prosłowiański,
niż wielu spośród nas, żyjących nad Wisłą, między Bugiem a Odrą.
Pozwalam sobie w zakończeniu zapewnić pana profesora o moim osobistym szacunku do Jego osoby, jako przykładu wyróżniającego się polskiego naukowca, pracującego poza granicami kraju i sławiącego dobre imię Polski. Będę miał zawsze
w pamięci Jego pomoc, jakiej udzielał młodym naukowcom, przybyłym z różnych
krajów na staż do Uniwersytetu Stanowego Wisconsin w Milwaukee w latach dziewięćdziesiątych, a z której to pomocy i wspierania niżej podpisany, jako ówczesny
skromny „scientific researcher” również nieco skorzystał, pracując w Centrum
Kompozytowym, kierowanym przez profesora Pradeepa K. Rohatgi’ego, uznawanego w świecie za „ojca kompozytów”…
Prof. dr hab. inż. Jerzy J. Sobczak
Dyrektor Instytutu Odlewnictwa
Kraków
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Foreword
On behalf of the Foundry Research Institute I heartily welcome all the participants
of the Polish-American Symposium honoring 75th birthday of our compatriot Professor Miłosz Piotr Wnuk, a prominent scientist of international reputation. According to the words of Professor Zenon Mróz of the Polish Academy of Sciences in
Warsaw “Professor Wnuk is a specialist on the world scale contributing his original
research into his area of expertise”. Around the world Professor Miłosz Wnuk is
better known as Michael Wnuk, a renowned American scientist. Today we will also
celebrate half a century of his teaching and research activities.
It is an honor and a privilege for me to serve as your host under the roof of this
Institute. I must admit that an event of the caliber of today’s Symposium does not
happen often, like the present Symposium, which we decided to name “Theoretical
Foundation of Excellence in Contemporary Engineering”.
Sometimes the practicing engineers look with a grain of salt on their colleagues,
who devote their lives to study the intricate complexities of Theoretical and Applied
Mechanics. Yet, all of us will agree that without a solid theoretical background and
without the basic foundation on which all of engineering rests, many of today’s discoveries, high-tech materials and technologies would not exist. Beginning with the
universal mind and inventor Leonardo da Vinci, followed by the great Galileo Galilei
and Sir Isaac Newton it was the Mechanics as the First Lady of the Court along
with Physics that faithfully served the Queen of all exact sciences – Mathematics –
that made possible such spectacular development of the engineering art, the design methods and the associated technologies. Without the mathematical foundations built over the past centuries and greatly influenced by ingenious individuals
such as Johannes Kepler, Gottfried Wilhelm Leibniz, Isaac Newton, Leonard Euler
and Albert Einstein, just to name a few, today we would not have so advanced
techniques and technologies that are available to the mankind and that are used to
enhance the quality and the standard of our life.
In certain circles Professor Miłosz Wnuk is viewed as a somewhat controversial but
colorful figure, which only adds to his personal charm. Without the doubt he is one
of the founding fathers of the 21st century Mechanics of Fracture and Damage Mechanics. For a number of years he has presented these subjects and his own contributions to these areas during his multiple lectures at the Institute of the Fundamental Problems of Mechanics in Warsaw and the AGH – University of Science and
Technology in Krakow. The Foundry Research Institute was also a host to Professor
Wnuk’s seminars and consulting services.
More detailed information on the activities and researches of Professor Miłosz
Wnuk is contained in the volume of the Monograph available to the participants of
the Symposium. There you will find tidbits of information related to His research
and teaching activities not only in the United States of America and Poland, but
also in the countries such as Great Britain, Germany, Russia, former Yugoslavia,
Italy, China and Ukraine. For the past several years Professor Wnuk serves as
a consultant to the National Aeronautics and Space Agency, NASA. His NASA-
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sponsored report related to the Cassini Mission to Saturn is reprinted in the Monograph.
Professor Wnuk has authored almost one hundred technical publications in refereed journals and several texts intended for graduate students and advanced engineers. Two editions of his college text “Foundations of Fracture Mechanics” were by
a text-book published in 2009 under the same title by Akapit Publishers associated
with the AGH – University of Science and Technology, Krakow. Professor Wnuk
wrote a major chapter and served as an editor for a book on “Nonlinear Fracture
Mechanics” published by Springer Verlag in 1990. He is a life member of the New
York Academy of Sciences and an associate member of the Philosophical Society of
Cambridge, England. Professor Wnuk has worked at Cambridge University in 1970
as a “Distinguished Visiting Scholar” in the Department of Applied Mathematics and
Theoretical Physics at the same college where once Sir Isaac Newton has worked.
Some essential theoretical results derived by Professor Wnuk bear His name, such
as “Wnuk-Rice-Sorensen equation”, “Wnuk-Knauss equation” or “Wnuk’s criterion”,
as indicated in the Monograph.
Finally I wish to express my heartfelt thanks to Professor Miłosz Wnuk for his active participation in the work needed to prepare this Symposium and for his willingness to travel the substantial distance between the remote city of Milwaukee in
Wisconsin and Kraków. Despite his over forty years long absence from Poland, Professor Wnuk frequently emphasizes that he feels more Polish than American. Indeed, sometimes he appears more pro-Polish and more pro-Slavic than all of us,
who live in this land between the rivers of Bug and Odra.
I take this opportunity to express my respect to this man, who serves Poland by
promoting its’ good name and its’ fame overseas and in Europe. I shall always remember His personal involvement and the help he rendered to the young scientists
from Poland, who happened to work at an American university. As a modest
“Scientific Researcher” I was one of those visiting scientists while working in the
Composites Center under the guidance of Professor Pradeep Rohatgi, known worldwide as “father of composites”…
Professor Jerzy J. Sobczak, DSc., PhD., Eng.
Director of the Foundry Research Institute
Krakow
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Curriculum Vitae
Miłosz Piotr Wnuk jest absolwentem Uniwersytetu Jagiellońskiego oraz Politechniki Krakowskiej. Na Uniwersytecie Jagiellońskim ukończył kierunek
fizyki teoretycznej w roku 1965, natomiast studia na Wydziale Mechanicznym Politechniki Krakowskiej kończył w 1959 roku; doktorat obronił w zakresie mechaniki teoretycznej i stosowanej w 1962 roku, a przewód habilitacyjny ukończył w czerwcu 1982 roku. Od roku 1959 pracował jako asystent w Katedrze Fizyki PK, a od roku 1963 do 1966 jako adiunkt.
W roku 1966 profesor M. Wnuk wyjechał do Stanów Zjednoczonych, gdzie
przez wiele lat pracował na różnych uczelniach, takich jak: California Institute of Technology w Pasadenie w Kalifornii, Uniwersytecie Stanforda,
Northwestern University, University of Michigan. Najdłużej był związany
z Uniwersytetem Wisconsin w Milwaukee, gdzie wykładał przedmioty ścisłe, między innymi matematykę stosowaną dla inżynierów i konstruktorów.
Od 1990 roku jest konsultantem NASA w Kalifornii, gdzie sporo uwagi poświęca astrofizyce i nawigacji kosmicznej.
W roku 1970 profesor Miłosz Wnuk pracował jako Distinguished Visting
Scholar w Departamencie Matematyki Stosowanej i Fizyki Teoretycznej
(Department of Applied Mathematics and Theoretical Physics, DAMTP) na
Uniwersytecie Cambridge w Anglii. Ta jednosemestralna wizyta była sponsorowana przez Marynarkę Wojenną USA, tzw. Office of Naval Research
(ONR) oraz przez British Science Council. DAMPT jest tym samym wydziałem Uniwersytetu Cambridge, na którym pracował Isaac Newton. Od roku
1976 Miłosz Wnuk zostaje pełnym profesorem w Stanach Zjednoczonych
na Uniwersytecie Stanowym Południowej Dakoty.
Jest autorem dwóch podręczników oraz ma na swoim koncie blisko sto publikacji naukowych. Materiały do tych publikacji zostały zebrane nie tylko
w USA, lecz także w wielu krajach Europy, takich jak Anglia, Niemcy, Włochy, Izrael, Jugosławia, Słowenia, Rosja i – oczywiście – Polska. Tutaj wykładał w Instytucie Podstawowych Problemów Techniki Polskiej Akademii
Nauk w Warszawie oraz na Akademii Górniczo-Hutniczej w Krakowie. Na
podłożu tego monograficznego wykładu dla doktorantów i pracowników
naukowych AGH, wygłoszonego w semestrze jesiennym w 1974 roku,
opracowany został Skrypt Uczelniany „Podstawy Mechaniki Pękania”, który
ukazał się drukiem w 1977 roku (pierwsze wydanie, Skrypt Nr 585) oraz
w 1982 roku, drugie wydanie, Skrypt Nr 822. Poprawiona i uwspółcześniona wersja tego skryptu została wydana w postaci książki pod tym samym
tytułem przez krakowskie Wydawnictwo Naukowe „Akapit”, związane
z Akademią Górniczo-Hutniczą. Wydanie książki miało miejsce w 2008 roku.
Profesor Zenon Mróz z Polskiej Akademii Nauk w Warszawie tak oto pisze
o tym podręczniku: „Po zapoznaniu się z konspektem podręcznika mogę
stwierdzić, że obecne ujęcie stanowi nowy podręcznik akademicki bardzo
potrzebny do prowadzenia wykładów i rozwoju badań w tej ważnej dzie-
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dzinie nauki i inżynierii. Autor jest uznanym specjalistą w skali światowej
wnoszącym znaczący oryginalny wkład do tej dziedziny badań”.
W roku 1981 profesor Wnuk stworzył w Belgradzie, jako stypendysta Fundacji Fulbrighta oraz Amerykańskiej Akademii Nauk w Waszyngtonie,
pierwszą Letnią Szkołę Mechaniki Zniszczenia Ciał Stałych na Bałkanach,
tzw. International Fracture Mechanics Summer School, w skrócie IFMASS.
Ostatnia taka szkoła, IFMASS10, miała miejsce w czerwcu 2008 w Zlatiborze w Serbii. Przez okres jednego roku akademickiego 1981/1982 profesor
Wnuk pracował na Uniwersytecie Belgradzkim. Wszystkie jego wykłady
na kolejnych Szkołach Letnich na Bałkanach zostały opublikowane w Proceedings IFMASS, wydawanych w Belgradzie. W 1991 roku profesor Wnuk
przyjął funkcję koordynatora i dyrektora zaawansowanego międzynarodowego kursu z Mechaniki i Fizyki Zniszczenia zorganizowanego pod patronatem Międzynarodowego Ośrodka Naukowego CISM w Udine, Italia.
W roku 1983 profesor Miłosz Wnuk wykładał w Chinach na Uniwersytecie
Wuhan. W 1992 roku otrzymał prestiżowe stypendium od rządu Izraela,
tzw. Lady Davies Scholarship – i przez okres jednego semestru wykładał
w elitarnej szkole inżynierskiej w Haifie, tak zwanym Technionie. W 2003
roku przez jeden semestr wykładał również w języku angielskim w Wyższej
Szkole Inżynierskiej przy Akademii Górniczo-Hutniczej w Krakowie. Poczynając od roku 1996 Miłosz Wnuk jest współzałożycielem Międzynarodowych Warsztatów z Mezomechaniki przy Instytucie Fizyki Wytrzymałości
Ciał Stałych Rosyjskiej Akademii Nauk w Tomsku na Syberii. Jest również
członkiem Rady Redakcyjnej czasopisma poświęconego mezomechanice
Physical Mesomechanics redagowanego w Tomsku pod kierunkiem profesora Wiktora Panina, a wydawanego w Holandii przez prestiżowe wydawnictwo Elsevier.
Profesor Wnuk jest członkiem naukowego stowarzyszenie Sigma Xi, Nowojorskiej Akademii Nauk oraz Towarzystwa Filozoficznego w Cambridge, Anglia. Od roku 1983 jest prezesem Amerykańskiej Fundacji Kultury Polskiej,
która zalicza się do jednej z bardziej aktywnych polonijnych organizacji
promujących kulturę Polski w Stanach Zjednoczonych. Od roku 2005 pracuje w Polsce, podczas swych częstych wizyt w kraju, jako Przewodniczący
Rady Fundacji dla Polskiej Fundacji Kultury Amerykańskiej imienia Ignacego Jana Paderewskiego z siedzibą w Krakowie.
18
Dr. Michael P. Wnuk
Biographical Information
Professor Michael P. Wnuk teaches Engineering Mechanics at the University of Wisconsin – Milwaukee. He has taught and performed research at
various schools in the United States, including Michigan State University,
Stanford University, California Institute of Technology and Northwestern.
Dr. Wnuk has also worked abroad in England, Poland (his native country),
Germany, Russia, Italy, Yugoslavia and China. In 1970 he worked as
a Distinguished Visiting Scholar in the Department of Applied Mathematics
and Theoretical Physics at the University of Cambridge, UK. The British
Science Council and the Office of Naval Research of the US have sponsored his research there. The other sponsors of his research include NATO,
NASA, the National Science Foundation, National Academy of Sciences and
the National Institute of Standards and Technology.
In 1991, he was appointed a Fulbright Scholar in Yugoslavia. He is one of
the founding fathers of the First Yugoslav Summer School in Fracture Mechanics, established in 1981 and co-sponsored by the National Academy of
Sciences, Washington, DC. In 1992, he received the Lady Davies Scholarship from the Government of Israel. He is a member of the Sigma Xi Research Society, an Associate Member of the Cambridge Philosophical Society in England, member of the American Academy of Mechanics, and
a life member of the New York Academy of Sciences. He wrote two books
and authored almost a hundred technical reports published in various international refereed journals.
Dr. Wnuk is one of the co-founders and a co-chairman of the International
Conference and Research Workshops on Mesomechanics, which convenes
every two years (in 1996, Tomsk, Siberia, in 1998, Tel Aviv, in 2000 in
China, and in 2002 in Denmark at the Aalborg University) in order to
merge interdisciplinary research of high-tech nature involving Physics at
nano-scale, Materials Engineering and Mechanics.
He has been selected an ASEE/NASA Summer Faculty several times; in
1966 at the Johnson Space Center – NASA White Sands Test Facility in
New Mexico, and then in 1998, 1999, 2000, 2001, 2002 and 2003 at California Institute of Technology/Jet Propulsion Laboratory in Pasadena, California.
Since 1987 Dr. Wnuk serves as President of the “Panslavia International
Research Institute, Inc.”, which assists multinational partners in trade,
science and technology transfer with particular emphasis on global problems of ecology and medical R&D. In 2005 Professor Michael Wnuk re-
19
ceived Ronald Reagan Golden Medal from the National Republican Congressional Committee as recognition for his active support at the international level of the policies of President George W. Bush.
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Anegdotycznie o Miłoszu Wnuku
Miłosz Piotr Wnuk
Nie wiem dlaczego Jacek Kozłowski, mój kolega z ławy szkolnej z czasu
pięcioletnich studiów na Politechnice Krakowskiej (1954–1959) ilekroć posyła do mnie maila, zamiast pisać w nagłówku moje imię tak jak to robią
wszyscy inni, otwiera swój tekst jednym słowem: „Wnuku”. Istotnie Jacek
jest chodzącą osobliwością i można powiedzieć, że jako taki posiada prawo
do posługiwania się własną licentia poetica. Inni nazywają mnie Miłoszem.
Mimo odmiennych poglądów na życie, bardzo się z Jackiem lubimy. Jacek
ma pewien rzadki dar: uwielbia Woltera, piękne kobiety i wysoki standard
życia. Jego poczucie humoru nie ma sobie równych, a pozytywne wibracje
energetyczne, które z siebie emanuje, dobrze wpływają na samopoczucie
osobników zgorzkniałych.
Oto kilka anegdot o Wnuku, dobrych czy też złych ale zawsze prawdziwych, spisanych przez Wnuka na prośbę Redakcji „Naszej Politechniki”.
Piszę sam po prostu dlatego, że nikt inny takiego zadania nie chciał się
podjąć. Czy to z braku talentu, czy też z innych ważnych powodów natury
politycznej, tego nie wiem. Kiedyś, wiele lat temu, Redaktor Kultury paryskiej, Jerzy Giedroyć, pozwolił sobie przetasować moje imię i nazwisko nazywając mnie w końcu „Wnukiem Miłosza”. Nie uwłacza to mojej godności
osobistej; wprost przeciwnie – nobilituje.
Ta zabawna gra słów sprzed lat została mi przypomniana tydzień temu
podczas uroczystości chrzcin w klasztorze Bernardynów w Alwernii koło
Krakowa, gdzie byłem gościem rodziców mamy niemowlaka, Maksymiliana. Piękny zabytkowy kościół z obrazem Matki Bożej Alwernijskiej przy
bocznym ołtarzu. Wzruszająca uroczystość przy dźwiękach organów, a potem rodzinne spotkanie przy suto zastawionym stole w restauracji „Alchemia”, też w Alwernii. I tam wydarzyła się taka historia (daję słowo, że nie
zmyślam). Pan Tadeusz, ojciec Agi, która urodziła syna, usiadł obok
i zwrócił się do mnie takimi słowami – „panie Miłoszu, wszyscy czytamy
i podziwiamy pańską poezję”. Próbowałem zwrócić jego uwagę, że mowa
chyba o Czesławie Miłoszu, laureacie nagrody Nobla. To on pisał wiersze.
Ale pan Tadeusz wiedział lepiej. Bynajmniej nie zbity z tropu, mój rozmówca ciągnął dalej „tak, rozumiem, pan jest zbyt skromny.” A potem dodał „zazwyczaj poeta musi umrzeć, aby stać się sławnym”. Poczem zaraz
dodał z podziwem „a pan nadal żyje”. Z takim dictum nie mogłem się nie
zgodzić. Nie chcąc komplikować sytuacji ani też psuć nastroju miłego wieczoru, przyznałem panu Tadeuszowi rację – „tak, żyję” powiedziałem, poczem oddałem mu jedną z moich wizytówek. Utwierdziło to tylko rozmówcę w jego przekonaniach. Ucieszył się bardzo i schował wizytówkę jak
święty obrazek. Powiedział też, że kiedyś odwiedzi mnie w Stanach.
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Otóż to, żyję jeszcze a już piszę o sobie anegdoty. Nie jest to całkiem dobrowolne. Poniekąd zmuszają mnie do pisania okoliczności oraz atmosfera
jaka powstała wokół tego pomysłu redaktora Naszej Politechniki. Niektórzy
uczeni w piśmie koledzy skomentowali ten zamysł mniej więcej tak: „po
przemyśleniu, nie chcę z tym mieć nic wspólnego”. Zabieram się zatem do
dzieła. Zadanie polega na tym – jak na wstępie zapowiedziałem – aby
przelać na papier kilka anegdot lub historyjek, wesołych lub smutnych, jakie zapewne każdy z nas posiada w swoim życiorysie. Spróbuję zatem
opowiedzieć to i owo o moich studiach na Politechnice Krakowskiej.
Zaczynałem studia w czasach kiedy dziekanem Wydziału Mechanicznego
był profesor Steindel, matematykę wykładali profesorowie Krzyżański i Barański oraz dynamiczna pani Majcherowa, natomiast wykład z Marksizmu
i Leninizmu prowadził Jan Betlej. Na starszych latach wykładowcami byli
perfekcyjny Janusz Walczak i jego wspaniały asystent Michał Życzkowski,
nieobliczalny Jan Korecki ze Lwowa oraz dystyngowani profesorowie Ilnickij i Riedel. Z tym ostatnim mam wspólne publikacje. Podczas wykładów
opowiadał nam jak podczas wojny sabotował produkcję dla niemieckiego
przemysłu zbrojeniowego. Uczył nas w ten sposób jak drobna na pozór
zmiana w doborze tolerancji może prowadzić do katastrofy. Byłem zauroczony studiami tak bardzo, że poważnie zaniedbałem swoje życie prywatne. Nawet zaproszenia na studniówkę w Liceum Jotejki nie przyjąłem od
pięknej panny Morstinównej zwanej „Murką”, i ta – z desperacji – wyszła
za mąż za chłopaka z kresów, Janusza Orkisza. Co prawda, po latach, Janusz przywiózł Murkę do mnie do Stanów.
A teraz ad rem. Czy wydarzyło się coś niezwykłego? Można by zacząć od
incydentu wybicia szyby w oknach domu zacnego profesora Webera, który
wykładał Części Maszyn na Wydziale Mechanicznym Politechniki. Szyba
została wybita podczas gry w piłkę z moim bratem Krzysztofem oraz kilkoma kolegami z ulicy Słonecznej, obecnie Bolesława Prusa, na Zwierzyńcu. Tam wówczas mieszkałem wraz z moimi rodzicami w Krakowie, bardzo
blisko Błoń. Ale to był przypadek banalny, który nie spowodował żadnych
poważnych konsekwencji poza tylko tym jednym, że od tej pory z profesorem Weberem łączyła mnie i mojego brata serdeczna przyjaźń. Gdy zakochałem się w Joannie Krajewskiej z ulicy Fałata, profesor Weber wymyślił
nadzwyczaj chytry fortel aby połączyć mnie z moją ukochaną Joanną –
niemal jak Zagłoba u Sienkiewicza. Mimo tych wszystkich zabiegów Joanna
wkrótce wyszła za mąż za hrabiego Pinińskiego z Warszawy, a ja wyjechałem daleko za morze – jak w ułańskiej piosence „O mój Rozmarynie”.
A oto bardziej już pikantna afera, która wywarła wpływ na niemal całe
je życie studenckie i zawodowe. Już na drugim roku studiów bardzo
upodobałem sobie fizykę i postanowiłem zapisać się do Koła Naukowego
Fizyki na PK. Kołem tym opiekowała się profesor Jadwiga Halaunbrenner,
żona szefa Katedry Fizyki profesora Michała Halaunbrennera. Wykład
z fizyki na naszym roku prowadził profesor Michał Halaunbrenner i ten
właśnie wykład stał się inspiracją do moich działań. Zaczęły się one dosyć
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niefortunnie. Tak przynajmniej mogło się wówczas wydawać. A naprawdę
jak to było?
Pani profesor Jadwiga Halaunbrenner, opiekun Koła Fizyków, podczas rozmowy ze mną zasugerowała, abym przygotował godzinny referat na temat
ograniczeń stosowalności Drugiej Zasady Termodynamiki w oparciu o pracę wybitnego polskiego fizyka Mariana Smoluchowskiego. Chodziło o tak
zwane procesy antyentropijne zachodzące w mikroskopowych obszarach,
gdzie widoczne są fluktuacje gęstości oraz energii cząsteczek. Oczywiście
cytowałem wyniki prac francuskiego fizyka doświadczalnego Perrina – dokładnie tak jak to zrobił wcześniej Smoluchowski i Einstein, kiedy opisywali
ruchy Browna. Na koniec, uzupełniłem ten wykład cytatem z Biblii dotyczącym śmierci termicznej wszechświata, która musi nastąpić w odległej
przyszłości, kiedy zgodnie z Drugą Zasadą znikną różnice temperatur
w kosmosie. Brak takich różnic oznaczać będzie, że entropia osiągnęła
maksimum i niemożliwy będzie jakikolwiek przepływ energii. Biblię pożyczyłem od księdza profesora, przyjaciela mojego ojca, i miałem ją pod ręką na wypadek pytań po referacie. Kiedy skończyłem, głos natychmiast
zabrała pani profesor Jadwiga Halaunbrenner. Był to głos pełen wyrzutu
i oburzenia. Oświadczyła między innymi, że treść mojego referatu nie była
z nią uzgodniona i że Koło nie ponosi odpowiedzialności na moje rewelacje
biblijne.
Zaraz potem stał się cud. Mąż pani profesor, Michał Halaunbrenner podniósł się z ławki i życzliwym tonem zaprzeczył wszystkiemu o czym właśnie
mówiła jego żona, a także dobrodusznie oświadczył, że referat był „doskonały” i „głęboko przemyślany”. Na tym dyskusja dobiegła końca (znacznie
później dowiedziałem się, że profesor Halaunbrenner był człowiekiem głęboko wierzącym). Ten epizod z Biblią miał być jednak dopiero początkiem
konfiktu, jaki wkrótce rozgorzał na dobre i który posiadał wszelkie znamiona „zimnej wojny” między mną a panią profesor Jadwigą Halaunbrenner.
Po obronie pracy magisterskiej w 1959 roku, której istotny wynik został
opantentowany w Polsce, wróciłem do Katedry Fizyki tym razem w charakterze asystenta. Braki w moim wykształceniu z fizyki jakoś szefa Katedry
nie zniechęcały do mnie. Zresztą, wkrótce miałem je nadrobić z nawiązką
kończąc pełne studia fizyki na Uniwersytecie Jagiellońskim. Jednak żadne
moje wysiłki i dobre oceny w indeksie z UJ nie przekonały pani profesor
Jadwigi Halaunbrenner co do moich kwalifikacji oraz podejścia do pracy
w wybranej przeze mnie dziedzinie. Już w pierwszym tygodniu mojego
stażu w Katedrze Fizyki upuściłem na podłogę w naszej pracowni kosztowny aparat Hoffmana służący do elektrolizy wody. Aparat rozbił się w drobny mak. Dla pani profesor był to nie tylko wstrząs, ale również niezbity
dowód, że praca w fizyce nie jest mi przeznaczona. A już na pewno nie
w fizyce doświadczalnej, w której profesor była prawdziwym mistrzem.
Jednakże jej mąż, szef Katedry Fizyki, do którego zwracała się – tak jak
wszyscy inni w Katedrze – per „panie profesorze”, był odmiennego zdania.
Kontrast był tak oczywisty, że wprost rzucał się w oczy nawet osobom postronnym. Różnica zdań między szefem a panią profesor była szczególnie
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widoczna podczas seminaryjnych wykładów (nie jeden z nich był wygłaszany przeze mnie) oraz zebrań organizacyjnych wszystkich pracowników
Katedry Fizyki. Odbywały się one – tak jak to ma miejsce również dzisiaj –
w pięknej nowej sali wykładowej na pierwszym piętrze zabytkowego budynku przy ulicy Podchorążych 1. Ilekroć przydzielano nowe zadania pracownikom naukowym Katedry, w tym również mnie, kierownik Katedry po
prostu nie przyjmował krytycznych uwag pani profesor Halaunbrenner, jakie od czasu do czasu wypowiadała pod moim adresem. Wydawało się, że
tej zimnej wojnie nie będzie końca.
I tak było do momentu, kiedy opatrzność sprawiła, że doszło do konfrontacji oraz zakończenia wojny w sposób tak spektakularny, że trzeba chyba
powołać się na naocznych świadków zdarzenia, które teraz opiszę. W pewien pogodny czwartek wczesnym popołudniem profesor Michał Halaunbrenner wezwał mnie do swojego gabinetu i polecił, abym zawiózł panią
profesor na ulicę Warszawską, gdzie jak wiadomo mieści się siedziba
główna Politechniki. Pomysł był szalony, ale taki również był nasz szef.
W owych czasach nikt w Katedrze, oprócz docenta Lepszego, nie posiadał
samochodu, a więc w grę wchodził tylko mój motocykl wschodnioniemieckiej marki „MZ” o pojemności silnika 250 centymetrów sześciennych. Nie wiem czy pani profesor kiedykolwiek w życiu podróżowała na
takim wehikule. Z pewną obawą pokazałem jej zatem jak usadowić się na
tylnym siodle. Motor zapalił i ruszyliśmy w drogę.
Dla tych czytelników, którzy znają topografię ulicy Podchorążych w Krakowie będzie oczywiste, że pierwszy skręt, jaki musi być wykonany jeśli pojazd zmierza na ulicę Kazimierza Wielkiego i dalej poprzez Łobzowską do
ulicy Szlak oraz Warszawskiej, jest skrętem w lewo. Na domiar złego, przy
wykonywaniu zakrętu należy tam przeciąć tory tramwajowe, a także –
zgodnie z przepisami drogowymi – należy oddać pierwszeństwo wszystkim
pojazdom zmierzającym wzdłuż ulicy Podchorążych z przeciwnego kierunku. Taki właśnie manewr skrętu został wówczas wykonany. Kiedy jednak
mój motocykl czekał aż pojazdy z naprzeciwka przejadą w kierunku Bronowic, nadjechał z tyłu tramwaj i w ostatniej chwili gwałtownie hamował.
Z rozpędu uderzył, co prawda z niezbyt wielką siłą, w nasz motocykl. Wiedziona instynktem samozachowawczym pani profesor rzuciła się rozpaczliwie na szybę kabiny tramwaju, a potem spiesznym krokiem wróciła do Katedry. Ja również tam wróciłem. Szkód wielkich nie było, więc policji nikt
nie wzywał. Nasze relacje o wypadku – przekazane profesorowi Michałowi
Halaunbrennerowi – trochę od siebie się różniły. Moja wersja była taka, że
obowiązkiem kierowcy jest zaczekać, właśnie stojąc na środku ulicy w poprzek torów tramwajowych, aż do momentu kiedy wszystkie pojazdy z naprzeciw przejadą. Pani profesor nie zaprzeczała kolejności wydarzeń, ani
też meritum całej sprawy, jednakże była święcie przekonana, że czekanie
na torach tramwajowych było niewłaściwe i prawdopodobnie było to celowe działanie aby spowodować wypadek i doprowadzić do cielesnego obrażenia lub też nawet śmierci osoby siedzącej na tylnym siodle.
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Pomimo lat, które od tej chwili upłynęły i pomimo głębokiej i szczerej przyjaźni która nas przez wiele lat łączyła i stosu listów które do siebie napisaliśmy, nie dałbym głowy, że nasza szanowana szefowa, pani profesor Jadwiga Halaunbrenner, nie była w głębi ducha przekonana, że całe opisane
wydarzenie było starannie zaaranżowanym zamachem na jej życie.
Ten niecodzienny wypadek przeszedł do historii Katedry Fizyki Politechniki
Krakowskiej, obecnie Wydziału Fizyki, Matematyki i Informatyki Stosowanej. Także moje liczne wycieczki motocyklem wzdłuż i wszerz całej Europy,
i to w czasach głębokiej komuny, należą już do historii. Podobnie jak moje
rozmowy z panem Morawskim w Paryżu, ambasadorem przedwojennego
Rządu RP na uchodźstwie w Londynie. To też już historia. I tak samo ma
się rzecz z moimi wizytami w Maison Laffitte koło Paryża, gdzie przebywałem u Redaktora naczelnego Kultury, Jerzego Giedroycia. Byłby to osobny
rozdział do napisania. Tam właśnie Józef Czapski, malarz i pisarz, ofiarował mi swą książkę o zesłaniu w głąb Rosji „Na nieludzkiej ziemi”. Od
Czapskiego dowiedziałem się, że w bardzo ciężkich chwilach w życiu, kiedy
wszystko zawiodło, od zwariowania ratowała go muzyka Chopina. Dobrze
o tym wiedzieć.
Mój odczyt w Kole Fizyków o granicach stosowalności Drugiej Zasady Termodynamki sprawił, że przez wiele lat wykładałem właśnie termodynamikę
na uniwersytecie. Sprawa z cytatem z Biblii znalazła ciekawy epilog. Rok
lub dwa po moim odczycie złożyłem w redakcji Czasopisma Technicznego
wydawanego na Politechnice Krakowskiej artykuł o tym samym tytule co
odczyt. Artykuł został poddany dwóm recenzjom i przekazany do drukarni.
Otrzymałem już nawet wstępnie wydrukowany tekst do korekty autorskiej.
I wtedy nieoczekiwanie wpłynęło polecenie, zdaje się że od profesora
Chrzanowskiego z Zakładu Maszyn Cieplnych, aby druk wstrzymać, bo tak
podobno życzyła sobie lokalna jednostka PZPR przy Politechnice. Rzecz
nigdy nie została wydrukowana, a szkoda. Był to przejrzyście napisany
manuskrypt, sporo matematyki, fizyki i... ani słowa o Biblii. Ale wtedy partia orzekała o tym co jest politycznie poprawne a co nie. Widocznie Druga
Zasada Termodynamiki naszym bonzom nie spodobała się. Bardzo podobne rzeczy działy się w tamtych latach w Związku Radzieckim, gdzie o poprawności nauki decydowała partia.
Co do „zamachu” na życie naszej surowej szefowej, to wspominam trochę
podobną, lecz bardziej tragiczną historię jaka wydarzyła się na Politechnice
Krakowskiej w latach siedemdziesiątych w związku z moim przewodem
habilitacyjnym. Przewód został otwarty na Politechnice w roku 1966, cztery lata po obronie pracy doktorskiej. Niestety był to również rok mojego
wyjazdu do USA na jednoroczne stypendium w Kalifornijskim Instytucie
Technologicznym, tak zwanym „Caltechu”. Po wielu tarapatach, w których
uczestniczyło kilku dziekanów a także rektorów i sekretarzy partyjnych,
przewód został uwieńczony pełnym sukcesem lecz dopiero w roku 1982,
a zatem w 16 lat po jego rozpoczęciu i w rok po ogłoszeniu stanu wojennego, kiedy Solidarność zaczęła dochodzić do głosu na scenie politycznej
Polski.
25
Atmosfera polityczna w PRL-u definitywnie nie sprzyjała nierozsądnym
osobnikom takim jak ja, którzy żyjąc w Stanach upierali się przy postawieniu na swoim i uzyskaniu stopnia naukowego, który de facto do niczego
nie był potrzebny. W Stanach pojęcie „habilitacja” nie istnieje, a w Rosji
natychmiast kojarzy się ze wzbudzającym podejrzenie terminem „rehabilitacja”. Jest to proces, któremu podlega osobnik wypuszczony z łagru. Ale
w zbiorowym umyśle narodu, ten co siedział, żeby nie wiem ile takiej rehabilitacji przechodził, zawsze będzie podejrzany.
To samo było w Polsce. Jeden z rektorów Politechniki Krakowskiej miał podobno powiedzieć „po moim trupie, jeśli Wnuk ukończy habilitację na Politechnice”. Okazało się, że miał rację. Zmarł tragicznie, a ja otrzymałem
dyplom „doktora habilitowanego” datowany 16 czerwca 1982 roku.
Kiedy dziś, późną jesienią 2009 roku, słucham przyjaznych wypowiedzi
premiera i prezydenta Rzeczpospolitej skierowanych pod adresem Joe Bidena, vice-prezydenta USA odwiedzającego Warszawę, nie mogę uwierzyć,
że kiedyś było inaczej. A było bardzo inaczej... aż się łza w oku kręci.
26
Michael P. Wnuk
75 years of life and 50 years of teaching and
research
12 September 2011
Dr. Michael P. Wnuk
College of Engineering and Applied Science
University of Wisconsin Milwaukee
Milwaukee, WI 53201
Phones: 414-962-0687 or 414-217-6665
Email: [email protected] or [email protected]
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Sample Technical Publication
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Kwantowa teoria propagacji
quasi-statycznych szczelin w ośrodkach
niesprężystych
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42
Mathematics of Cassini’s Journey
to Saturn (1997–2004)
Michael P. Wnuk
NASA Visiting Scientist at JPL/Caltech
ASEE/NASA Summer Faculty at JPL/Caltech
Permanent affiliation: University of Wisconsin –
– Milwaukee
July 2000
CONTENTS
1. Introduction
- An Overview of Cassini Mission
Cassini as a Link Between Newton’s Orbital Mechanics and the
Space Exploration Program in 21st Century
2. Numbers, Functions and Operators
Numbers
Functions
Operators
Differential Equations
3. Calculus Underlying Orbital Mechanics
Motion in the central force field
Orbits of planets and spaceships
Navigating the Spaceship
4. Scalars, Vectors, Quaternions, Matrices and Tensors
Scalars that describe Cassini mission
Vectors and quaternions pertinent to the mission
Matrices and tensors applicable to the mission
43
1. Introduction
At the onset of the 21st century, in July of the year 2004, a man-made device will be landing on the cold surface of Titan, one of the moons of Saturn. It is a giant moon by all standards, it is bigger than a planet Mercury
and so, it was able to retain its own atmosphere, lakes and maybe even
oceans of methane and ethane. Of course, it pales in size compared to the
planet Saturn, which holds Titan permanently in the grasp of its gravitational field. The Cassini spacecraft that shall deliver this device, called the
Huygens probe, to the surface of Titan is an American built spaceship, designed and launched by NASA. One substantial section of the NASA’s Jet
Propulsion Laboratory in Pasadena, California is involved in navigating and
controlling the Cassini ship as it hurdles into the deep space further and
further away from Earth heading toward its final destination – Saturn,
which is not to be reached until July of 2004.
The mission began on October 15 of 1997 when the Cassini ship weighing
some 5,600 kilograms was launched into space by the most powerful US
Air Force rocket, Titan IV/Centaur, out of Cape Canaveral. The launch was
flawless, and it provided enough acceleration to the spacecraft so that the
“escape velocity” was exceeded allowing the ship to leave the gravitational
field of Earth and sail toward the vast space of the planetary system. During the initial stage of the journey (1997 - 1999) the trajectory of the Cassini’s flight seemed to be a bit convoluted, as it involved a lot of backtracking and sudden twists in otherwise regular elliptical orbits. These
strange maneuvers were motivated by a desire to get a “free ride”, at
least some of the way. Such free riding is made possible when the gravitational pull of the large celestial bodies, such as Earth, Venus and Jupiter is
put to work to accelerate the ship by a rapid variation of its orbit. Each
time such a sudden change in the trajectory of the spacecraft occurs, the
velocity is increased at no expense of the rocket fuel. Operations of this
type are known in the Orbital Mechanics as “gravitational assist” maneuvers. Four of those are planned for the Cassini, two involving Venus, then
Earth and the last one is to be carried out in the vicinity of Jupiter, see the
Figures ii, iii and iv. Total savings on fuel amounts to 75 tons of rocket
fuel, not an insignificant number.
The most recent swing-by of the Cassini spaceship required a close encounter with the planet Earth. In fact, Cassini’s altitude during this swingby operation was only 1176 km above the surface of Earth, which is considerably less than the altitude of geostationary satellites that orbit Earth at
42,164 km above the sea level. The speed of the Cassini spaceship,
though, equaled then 19.03 km/sec, which exceeded the escape velocity
from this particular height (10.29 km/sec) by a factor of 1.85. This provided an ample margin of safety, and is something to think about for
people who tend to worry about objects from space coming into close
proximity with our planet.
44
This event took place on August 18 of 1999. The “slinging action” of Earth
gravitational field did not fail, and Cassini is now on its way to Jupiter,
where the last gravity assisted boost of the ship velocity will be executed
on December 30, 2000, see Figure ii. After that, sometime around July 1
of 2004 the ship will approach its destination site, the majestic planet of
Saturn, hugged by the set of rings, consisting of zillions of the tiny frozen
rocks that like miniature moons circle around the planet, all positioned in
one plane. Ship engines will then be turned on to slow it down and maneuver through the gap in the rings to get the ship into the position suitable for expelling the Hyugens probe in the direction of Titan. It will take 3
weeks for the probe released from Cassini to descend and land (hopefully
not crash) on the surface of Titan. This giant moon resembles the primordial Earth, as it existed billions of years ago.
Whatever is learned from this frontier exploration mission to Titan may
help to solve the mysteries of our own past, speaking in the astrophysical
time units, of course. Thus, we may learn how did Earth develop from its
early stages of growth to what it is today: an extraordinary life-supporting
planet that by all standards appears to be unique in the vast and hostile
universe that surrounds us. Better understanding of the mechanism by
which the icy rings of Saturn were formed will help the astrophysicists to
revise their theories about the origin and evolution of galaxies and galaxy
clusters. And this is an important chapter in the history of the Universe.
To put it briefly, the information and bits of data transmitted back to Earth
from the Cassini spaceship will influence not only the scientific thinking (as
it already does), but it may alter certain technologies, especially in computer sciences and communications, currently in use on Earth. Therefore,
the two primary aspects of the Cassini mission, the fundamental and the
applied one, are so intricately interconnected that no one can discern
a clear boundary between them.
Data collected by the Hyugens probe will be transmitted back to Earth to
be intercepted by the three large radio-antennae positioned in a triangle
involving three continents: Australia, Spain and California. The spacecraft
itself will have sufficient amount of power (generated by its onboard radioactive electrical device) for the period of 4 years. The research and the
data to be collected are focused, among other things, on the nature of the
magnetic field of Saturn. This field is of a peculiar shape, greatly distorted
and so uniquely asymmetric that there is nothing like it in the entire solar
system. After 2008 the transmission of data via radio signals will cease
(formally the mission ends in 2008), but the spacecraft Cassini will forever
remain within the hold of the powerful gravitational field of Saturn. A mere
speck of dust in the sky, circulating an intricate path of elliptical orbits
without end… and yet, a wonderful proof of the presence of some smart
thinking species in the Universe, us.
And, since it took 33 states within the United States and 19 nations in Europe to participate in the preliminary stages of the project, the Cassini
45
mission may truly be said to represent the best of what today’s science
and technology – measured on the global scale – can do.
Continuity of human thought is remarkable. What began in 1655 with the
observation of the Saturn’s rings by the Dutch astronomer Christian Hyugens, and was followed by about two decades later by discoveries of
French-Italian astronomer Jean-Dominique Cassini (who found four of the
Saturn’s moons), will now be used to guide the spaceship Cassini through
the gap in the rings plane and eventually land the Hyugens probe on the
surface of Titan, a good observation point to take a closer look at this
spectacular planet. For science, an almost four centuries of elapsed time
that separate the past and the present discoveries appear to have no significant meaning.
When the cornerstone of all Mechanics (including Celestial Mechanics),
Newton’s awesome book of “Principia” appeared in print, the author of the
book was asked how was he able to determine with mathematical precision the laws that so absolutely guide remote celestial objects in their orbits. Newton’s answer was “I stood on the shoulders of the giants”. What
he meant by this were the discoveries, which preceded him, those of
Polish astronomer, Nicholas Copernicus, a bold proponent of the heliocentric system, and those of a German scientist, Johannes Keppler, who modified Copernicus’ circular orbits to the elliptical ones. These elliptical orbits
were then confirmed mathematically by Newton’s Universal Law of Gravity
– one law for the entire Universe. Excluding extremely minor corrections
arising from the Einstein’s General Theory of Relativity, these findings of
the past “giants” stand the test of time.
Figure “i” illustrates an astonishing evolution in the planetary science since
1676, when Cassini drew a sketch of Saturn, till 1981, when Voyager 2
sent to the NASA base on Earth a picture of Saturn, shown in the figure.
The next step will be a close-up view of Saturn that we will receive in July
2004 when Cassini arrives at its destination. The progress of the mission is
documented by the Figures ii, iii and iv.
46
Figure i. The original sketch drawn by Jean-Dominique Cassini in 1676
(top), and the picture of Saturn received via radio signal from Voyager 2
in 1981 (bottom).
47
Figure ii. Outline of Cassini’s trajectory for the entire journey from Earth
to Saturn (October 15, 1997 - July 1, 2004).
Four gravitational assist maneuvers, which require a “close encounter”
with another planet, such as Venus (twice), Earth and Jupiter, are shown
in this diagram.
Figure iii. Our own planet and the Cassini spaceship during the “swingby”
maneuver on August 18, 1999.
48
Figure iv. Closest approach to Venus for a gravitational assist maneuver
on April 26, 1998.
2. Numbers, functions and operators.
Numbers
We all know what numbers are. Today, even toddlers can count. I am told
that they loose this ability, though, somewhere in high school when they
are introduced to the computers. Relations between numbers are often
expressed with aid of the algebraic equations, in which symbols, such as
“a”, “b”, “c” or “x” are used to designate a certain number. You may not
remember all different names of groups of numbers, which may be referred to as natural, rational, irrational, decimals, real, imaginary, complex, quaternions – and what else do we have? They are all numbers, no
more complicated then a lengthy string of numerals that makes up our
Social Security number or your checking account number.
It is the manipulations performed on these numbers that are of interest
to us. To begin to manipulate we need to introduce the notion of a function.
49
Functions
Let us begin with a simple demonstration, in which an everyday situation
will be interpreted in a scientific way. Suppose, you inserted 75 cents into
a vending machine, and then made your selection by pressing a button
“Coca Cola”. Let’s reiterate: 75 cents go in, a can of coke comes out. This,
in fact, is what all functions do: they transform the data at the input
(a number) into the result obtained at the output (another number). To
make our vending machine comply with this strict definition, the coke
would have to have a certain number assigned to it; but this does not alter
the general principle. The numbers used for input are said to belong to
a set named domain, while the outcomes belong to a different set, called
range.
The easiest way to think of a function is to visualize a black box equipped
with an input, an output and labeled with some sort of a suitable name;
the name is necessary to distinguish one function from another. If we depart from the vending machine and venture into the realm of numbers, we
may suggest the following designations for the three elementary functions:
f 2 = sin(x)
f 3 (x) = exp(x)
f 1 (x) = x2
Now, we are using symbols (instead of objects), but the meaning is the
same. To demonstrate the action of these four functions (the fourth one is
our vending machine, don’t forget it), we have collected all examples in
Fig.1a, and – by using the black box concept, we have shown how all of
these functions work. This should be rather obvious, but to make it even
more obvious we have collected the pertinent data in Table 1.
Table 1
Function
f l (x)
:
input
output
1
2
3
1
4
9
f 2 (x):
input
output
0
π/4
π/2
0
.7071
1
f 3 (x):
input
output
0
1
2
1
2.7183
7.3891
In fact, almost all phenomena that are seen, experienced or measured can
be represented by one function or another. Functions results from reasoning, which is either elementary (well, what’s “elementary” for the master
Sherlock Holmes, may not always be so for dear Dr. Watson), or they
arise from Calculus. And, what is this benevolent monster, the Calculus?
To answer this question we need to learn a little bit about manipulating
functions, such as transforming one into another and the like. This proposition resembles somewhat a dirty trick common to all wicked witches;
changing a prince into an ugly frog, or vice versa. That’s exactly what we
propose to do in the following sub-section with some help from our mathematical magic wand – an operator.
50
Operators
Calculus implies use of operators and these are intimately related to the
differential equations. What are these mathematical contraptions? First
of all, they are perfectly innocent, since they do no harm, just transform
one entity into another. In most cases by “entity” we mean a function –
but not necessarily. When you travel abroad and visit several countries,
you will have to convert your home currency, say dollars, into foreign currencies such as pounds, zloties, marks, tolars, liras, francs, rubles etc.
This simple feat is accomplished at the local bank or an exchange post that
accepts the US dollars and produces equivalent amount of other currency.
Symbolically, we can describe these operations as follows. With M denoting a money-changing operator, we can write the following statements:
M GB [$]
M DE [$]
M I [$$]
M RU [$]
=
=
=
=
English pounds
German marks
Italian liras
Russian rubles
M PL [$] = Polish zloties
M SL [$] = Slovenian tolars
M F [$$] = French francs
M MX [$] = Mexican dollars
Note that the subscripts used on each “M” comply with the official designation of the country, for which the operator M applies. In a very analogous
way Table 2 shows several mathematical operators in action; they are
both linear operators, L, and nonlinear operators, N. Symbols “x” and “s”
are used to designate variables, while “a” is a constant.
Table 2
d
=
sin ( x )  cos ( x ) L−dif1=
f 
cos ( x )  sin ( x )
dx 
d
1
Ldiff
=
exp ( x )  =
exp ( x )  exp ( x ) L−dif=
f 
exp ( x )  exp ( x )
dx 
Ldiff
=
sin ( x ) 
Lint sin ( x )  =
− cos ( x )
∫ sin ( x ) dx =
L−int1  − cos ( x )  =
sin ( x )
as
 as 
1
Llaplace sin ( ax )  =
L−laplace
=
 a 2 + s 2  sin ( ax )
a2 + s2
2
d

N=
sin ( x )  cos 2 ( x )
diff 
sin ( x )  =
 dx

1
N int =
sin ( x )  ∫ e x sin
=
( x ) dx sin ( x ) − cos ( x )  exp ( x )
2
2
d y
N  y ( x )  =
exp ( x ) + 4sin exp ( x ) 
+ 4sin  y ( x )  y =exp x =
( )
dx 2
In this Table we have used classic examples of what becomes “bread and
butter” to a professional mathematician. Symbols “L” are reserved for linear operators, while “N’s” are used to denote the nonlinear operation.
And, finally, the superscript “-1” designates an inverse operation. This you
can easily verify by reading the lines containing L-letters backwards. If
there is no mistakes in Table 2, the reverse reading should produce the
same result as the L-1 operators do. For example, the inverse Laplace operator L laplace -1 applied to the fraction as/(a2 + s2) produces the original
51
function sin(ax), as expected. We see a pattern emerging from this selection of various operators. Each time an operator is put to work, it transforms one function into some other function (which is not much different
from the trick of a wicked witch, as described above). When you see symbols such as d or [ ]dx , don’t worry. If they are not in your everyday vo∫
dx
cabulary, just accept them as various “road signs” used in Calculus. Fig.
1b uses the concept of a black box to illustrate action of the Laplace and
the inverse Laplace operators.
Figure 1. Black box representation of the action of a function (a), and an
operator (b).
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Figure 1c. Concept of an operator, W, and an inverse operator, W–1.
Differential Equations
Since all phenomena encountered in the physical world involve certain
numerical entities – such as functions – and these are usually not independent variables, but they enter into numerous less or more intricate relationships, we must somehow cope with this state of affairs. To “cope” in
this context requires writing a differential equation, which is capable of
recording symbolically these various relationships, often dictated by the
Laws of Nature rather than a whimsical fantasy of a pure mathematician
(but these cases are not excluded, either). To do the right bookkeeping of
all the changes and transformations performed on the various functions,
we need – first of all – operators, and then, we need the differential equations that add precision to keeping track of these changes. To state this
briefly, we are ready now to enter the world of operators and functions
(far in the background, of course, will remain our good friends – numbers). A great Swiss mathematician, Euler, once said that nothing is quite
complete, until a number is generated, and he was right. Here, we are
playing a high-level game of functions and differential equations, but at
the end of the game we want answers. And these come – more often than
not – in form of numbers. A classic analogy can be given here: when the
slide-rules were replaced by the hand-held calculators, engineers used to
keep both a slide-rule and a calculator handy, just in case when some
doubtful results had to be double-checked. When the first generation of
computers showed up, the skeptical scientists kept their calculators handy
(and they still do!). And, so it goes on. When we are through with a complicated boundary value problem (to be discussed in a little while) and
whip some functions out of the fancy differential equations, we always
check how reasonable the results are by… looking at the generated numbers.
To bring this point home, let us take a look at the ubiquitous exponential
function, f(x) = exp(x). It can be generated in many different ways. Since
it is common in nature, it is used to describe population growth, stock exchange variations, radioactive decay, nuclear reaction processes, learning
and artificial intelligence and innumerable other phenomena, whether they
are designed by man or arise from nature. The classic example, quoted in
many textbooks, relates to the population growth and it goes somewhat
like this. Imagine starting a colony (of humans or some other species) on
an inhabited island. At the start you have N people, say the survivors of
a shipwreck. We assume that both sexes are present and that there no
predators present, while the food and energy are plentiful. You would now
expect that the rate, at which this population will grow, should be proportional to the current size of the population in question. Switching to Calculus, we reduce this lengthy sentence to a simple statement:
Rate is proportional to number N, or better yet
dN/dt = k
(1)
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Here, t denotes time and k is a certain constant, value of which is not essential to our considerations, and it may be set to be equal one (or anything else you want). In fact, this proportionality constant in a large
measure depends on the mood of two partners of the opposite sex, who
may desire to make love with an intention to produce an offspring. With
k = 1, Eq. (1) is re-written as
dN/dt – N = 0
(2)
subject to an additional (initial) condition that states N=N 0 at time equals
zero. Of course, we expect N to be a certain function of time, say N =
N(t). Equation (2) is an example of a linear differential equation (DE) of
the first order, the simplest one in the book. We notice that the operator
d/dx[ ] is listed in Table 2 as L diff . Thus Eq. (2) can also be expressed in
the form
L diff [N(t)] = N(t)
(3)
This equation contains an unknown function, N(t). What is this function?
Well, if you look to the left column listing various operators in Table 2, the
only differential operator that converts a function into itself is listed in the
second position from the top, and the function is (note that the independent variable “x” has been replaced by time t):
F(t) = exp(t)
(4)
When the initial condition is applied, stating that at time zero we have
started a colony containing N 0 individuals, the function (4) can be adjusted
to provide the solution to our problem defined by the differential equation
(1) or (2), namely
N(t)
=
N0
exp(t)
(5)
It is best if time t is expressed as a ratio t/T, where T is a characteristic
time interval, such as 9 months for the humans (on this planet). Then our
solution acquires the form
N(t)
=
N0
exp(t/T)
(6)
The outcome is shown in Fig. 2.
54
200
200
180
160
140
120
f ( λ)
100
80
60
40
20
10
0
0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
λ
0
2.7
3
3
Figure 2. Population increase over three characteristic time intervals according to Eq. (6). Note that the starting number was 10, while the characteristic time T = 9 months.
Functions are not only useful, but they are also fun to work with. Let us
switch the gears and talk Dynamics for a brief while (this will bring us
closer to the main topic: the Cassini mission). Look at Figure 3. If all
forces acting on the block of mass m are accounted for, such the viscous
force, cdx/dt, the elastic resistance of the spring, kx, and the inertial
force, md2x/dt2, the following relation emerges
Figure 3. Block of mass m suspended on a spring and a viscous damper
and set into a vibratory motion.
m
d 2x
dx
+ c + kx =
Fext ( t )
2
dt
dt
(7)
Of course, all of the above complies with the 2nd Law of Newton
Force = mass x acceleration
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Symbol F ext denotes an external force applied to the center of mass of the
block m. Setting the externally applied force F ext (t) to zero (for simplicity)
and assuming these constants:
m = 2, c = 1 and k = 8 (all quantities must be expressed in the correct
units!), we obtain
d 2x
dx
+ 0.5 + 4 x =
0
2
dt
dt
(8)
Since this is a second order linear differential equation, we need to specify
how the motion of the block “m” was started. To do this, the initial position x(0) and the initial velocity v(0) must be given (we note that the velocity and the position are related by the differential operator,
v(t)=dx(t)/dt). Let us choose the following initial values:
x(0) = 0 , and v(0) = 1
(8a)
The solution is not difficult to find, and it reads
x(t) = 0.5040 exp(-t/4) sin(1.9843 t)
(9)
This formula describes precisely the position of the center of mass of the
moving body at any given time. As can be seen from Fig. 4 the vibratory
motion is neatly described by a wave-like function diminishing in time due
to the presence of the viscous damping (the mid-term on the left side of
Eq. (8)). First derivative of x(t) yields the velocity, namely
v(t) = dx(t)/dt = [0.5040 sin(1.9843t) + cos(1.9843t)] exp(-t/4)
(10)
The corresponding graph is shown in Fig. 5. When studying stability of
such vibratory systems, it is often useful to eliminate time between equations (9) and (10). This yields a relationship between the coordinate x and
velocity v, as illustrated in Fig. 6. This type of diagram is known in Dynamics as “phase diagram”, and it is often very helpful in establishing the
range of system parameters for which the system remains stable. Contrary statement is also true: if occurrence of instabilities is detected theoretically, then the unstable and uncontrollable behaviors - or, catastrophes
– can be avoided. Such is the power of Calculus. Example of a transition
from an orderly motion (stable) to a chaotic one (unstable) is illustrated
by the Viewgraphs 1, 2 and 3.the unstable and uncontrollable behaviors –
or, catastrophes – can be avoided. Such is the power of Calculus.
56
0.416
0.6
0.5
0.4
0.3
0.2
0.1
X ( t)
0
0.1
0.2
0.3
− 0.28 0.4
0
1
2
3
4
5
0
6
7
8
9
10
t
10
Figure 4. A decaying wave-form is the solution of the differential equation
(8) subject to the initial conditions x(0) = 0 and v(0) = 1. Note that the
wave is contained within an exponentially decreasing envelope.
1
1.5
1.25
1
0.75
0.5
V( t)
0.25
0
0.25
0.5
0.75
− 0.695
1
0
1
0
2
3
4
5
t
6
7
8
9
10
10
Figure 5. Velocity of the vibratory system consistent with the solution of
the differential equation (8), shown here as a function of time.
57
1
1.5
1.25
1
0.75
0.5
V( t)
0.25
0
0.25
0.5
0.75
− 0.695
1
0.4
− 0.28
0.3
0.2
0.1
0
0.1
0.2
X ( t)
0.3
0.4
0.5
0.6
0.416
Figure 6a. When time is eliminated from Eqs. (9) and (10), the velocity of
the moving body can plotted directly as a function of its position. This type
of diagram, called a “phase-diagram”, is helpful in the studies of the stability of dynamic systems.
Figure 6b. Example of a phase diagram for a pendulum with arbitrarily
large angular amplitudes. The system is stable, as seen from the regular
shape of the diminishing spiral.
To conclude this section, let us modify somewhat the left hand side of Eq.
(7) by deleting the term with the first derivative (damping term), and by
adding the external force, the so-called the driving force exerted by an external agent, say F ext (t) = exp(-t). Now, we are seeking a solution to the
differential equation
d2
x ( t ) + 4 x ( t ) =exp ( −t )
dt 2
58
(11)
We may proceed to solve it in two different ways. One involves application
of the initial conditions, just like in the case discussed above. Let us
choose the initial position and the initial velocity as
x (0) = 1
x(0) = 0
x = dx/dt
(12)
An alternative way to approach solution to a second order differential equation is to specify the values of the function at two distinct points, usually
the end points of the chosen time interval. Let us study the motion of our
system for 10 seconds, which sets the time interval at 0 ≤ t ≤ 10, and let us
require that the function x(t) satisfies the following boundary conditions
x(0) = 1
,
x(10) = 0.5
(13)
As is seen from Fig. 7, the two solutions look substantially different despite the fact that they have been obtained from the same differential equation, Eq. (11). The corresponding solutions read
x 1 (t) = 0.6sin(2t) – 0.2cos(2t) + 0.2exp(-t)
(14)
for the initial value problem, and
x 2 (t) = 0.19sin(2t) + 0.8cos(2t) + 0.2exp(-t)
(15)
for the boundary value problem. The differences between the two solutions
can be appreciated by examining Fig. 7.
)
1.005
1.5
1.25
1
0.75
0.5
x1( t)
x2( t)
0.25
0
0.25
0.5
0.75
− 0.821
1
0
1
0
2
3
4
5
t
6
7
8
9
10
10
Figure 7. Two functions are shown x 1 (t) and x 2 (t). They resulted as the
solutions to the initial value problem and the boundary values problem,
respectively. Note that they both satisfy the second order differential equation (8).
59
VIEWGRAPH 1
Phase diagram for a dynamic system consisting of a nonlinear pendulum
subjected to viscous damping ρ, and governed by the following nonlinear
differential equation of the second order: d2θ/dt2 = -sinθ- ρdθ/dt. The
graph is “well-behaved” and there is no indication of any instabilities or
chaotic behavior.
VIEWGRAPH 2
When the differential equation in the Viewgraph 1 is altered by multiplying
the term sinθ by the time-dependent factor (f sint), in which f denotes
60
tensity of the coupling between the pendulum and an externally applied
electromagnetic field, the resulting phase diagrams fall into two distinct
categories, those associated with (a) order, and those associated with (b)
chaos. This viewgraphs shows the phase diagrams when we deal with
chaos, i.e., when the amplitude f is greater than the critical value of 1.87.
Yet, in this totally chaotic type of motion, it is possible to find order at
a deeper level (as revealed by the existence of an attractor, see the next
Viewgraph).
VIEWGRAPH 3
The set of points shown here represent 100,000 points within the phasespace, which correspond to successive values of coordinate (θ) and velocity (dθ/dt) recorded at the end of each successive cycle of a swinging pendulum, coupled with an electromagnetic field.
For a particular choice of the damping parameter ρ and the amplitude f,
the pendulum “goes crazy”, entering into the chaotic motion, as shown in
the previous Viewgraph. Existence of the attractor, though, is indicative of
the certain rules that apply to this chaotic motion. Attractor shown here is
an example of a Poincare section, which has a fractal dimension of 2.52.
In real life things rarely depend on just one variable. There may be two,
three, four or a lot more variables. When one considers problems in spacetime reference frame (such as designing a trajectory of a spacecraft, for
example), one needs four variables for a general case. Then, of course,
ordinary differential equations must be replaced by the partial differential
equations. In this event, the wealth of possible solutions, which comply
with the governing equations and satisfy the prescribed initial or boundary
conditions, is awesome. To make things even more intriguing the Nature
has kept in store a surprise for all of us who believed in a very regular
61
well-behaved Universe. It turns out that certain dynamical systems (like
a system of three bodies interacting with each other via gravitational field)
may, under certain conditions, become chaotic and totally unpredictable.
Such transition from order to chaos is illustrated for a case of a nonlinear
pendulum coupled with an electromagnetic field by the set of Viewgraphs
1, 2 and 3. We can only hope that our spaceships – out there in the deep
space – will not encounter field configurations that would cause an onset
of such strange chaotic behavior.
3. Calculus Underlying the Orbital Mechanics
Newton invented the Calculus and an invention of such a caliber would be
considered by many an achievement worth living one’s life. But, the spirit
of an experimental scientist has pushed him beyond this discovery. He put
his “magic” mathematical tool, the Calculus, to a test of cosmic proportions. He attempted to explain theoretically the existing data on the motion of the celestial bodies, such as planets in the Solar System.
How does gravity work? The experimental data suggested that the gravitational pull of Earth, or the gravitational acceleration is inversely proportional to the distance “r” measured between the centers of mass of the
two bodies exchanging the pull, say Earth and a satellite (such as the nature made old good Moon). For a so-called central-force problem, in which
the acceleration of a point is directed toward a given point, the pertinent
relations can be conveniently stated in polar coordinates (r,θ), as shown in
Fig. 8. The symbols e θ and e r , seen in Fig. 8, denote the unit vectors
(more about this subject is provided in the next section) – aligned with
axes θ and r, respectively. Using the polar coordinates Newton calculated
first the velocity of an object moving in the gravitational field, namely


r ( t + ∆t ) − r ( t ) dr 
dθ 

v=
=
er + r
eθ
( t ) ∆lim
t →0
∆t
dt
dt
(16)
and subjected this expression to the operator d/dt. In this way he was
able to calculate the rate of change for the velocity, which represents the
acceleration vector
62
Figure 8. Polar coordinates used to describe motion under central force
condition. Quantities shown in (a) are used to define the initial conditions,
while (b) shows two unit vectors aligned with the radial and transverse
axes, e r and e θ .
Figure 9. Four different types of the conic section: 1) circle, 2) ellipse,
3) parabola (infinite trajectory), and 4) hyperbola (infinite trajectory).
d  dr 
dθ  

a (t )
er + r
eθ
=

dt  dt
dt 
(17)
Combining the rules of Calculus with the geometrical considerations, he
also knew that

deθ
dθ 
= −
er
dt
dt
(18)
Then, Newton has evaluated the expression (17), reducing it to this rather
lengthy formula
63
2
2
 d 2r
dr dθ  

 dθ     d θ
a ( t ) = 2 − r 
eθ
  er +  r 2 + 2
dt dt 
 dt  
 dt
 dt
(19)
For the gravitational field of Earth the acceleration is
−g
RE2 
er
r2
(20)
in which both constants, g and R E are known; g = 9.81 m/s2, while the
radius of Earth R E equals 6370 km. Now, comparing expressions (19) and
(20) Newton arrived at two differential equations
d 2r
gRE2
 dθ 
r
−
=
−


dt 2
r2
 dt 
d 2θ
dr dθ
r 2 +2
=
0
dt
dt dt
2
(21)
A system of DE such as this one implies that we are searching for two solutions θ = θ(t) and r = r(t). As it turns out these two equations can be
reduced to a single second order differential equation, which involves
a new function u = u(θ) – defined as the reciprocal of r(θ), namely
1
d 2u
gRE2
, u (θ )
u
=
+
=
2
2 2
dθ
r0 v0
r (θ )
(22)
Note that the entities dθ/dt and dr/dt are eliminated via these relations
dθ
= r0 v0u 2
dt
dr
du
= −r0 v0
dt
dθ
(22a)
For the interpretation of the initial radius r 0 and velocity v 0 , you may refer
to Fig. 8. Now, equation (22) looks almost like the one solved in the preceding section, cf. Eq. (11). In fact, it is a little easier to solve than Eq.
(11), because on the right side of it we have just a constant rather than
the function exp(-t). The general solution of this equation is
u ( t ) = A sin θ + B cos θ +
gRE2
r02 v02
(23)
where A and B are constants to be determined from the initial conditions.
When θ = 0, u = 1/r 0 (see Fig. 8), and also when θ = 0, the radial component of velocity dr/dt = 0. This bit of information combined with the
second equation in (23) implies du/dθ = 0. From these two initial conditions, we obtain
64
1 gR 2 1
1
A=
0, B = − 2 E2 = −
r0 r0 v0 r0 1 − ε
(24)
For convenience we have used here the following notation
ro2 vo2
= 1+ ε
gRE2
(24a)
When these values are substituted into Eq. (23), we arrive at the solution
1
1
u (θ ) =
 −
 r0 1 + ε

1
 cos θ +
1+ ε

(25)
Since u(θ) = 1/r(θ), it is a simple matter to rewrite Eq. (25) in this final
form
r (θ ) =
r0 (1 + ε )
1 + ε cos θ
(26)
Here, the constant ε = [(r 0 2v 0 2/gR E 2) – 1], acquires the meaning of the
eccentricity of the satellite orbit, while the orbit itself – as defined by Eq.
(26) – represents a conic section, which can be described by one of the
three possible choices:
–
elliptical orbit, 0 ≤ ε ≤ 1 If “a” and “b” denote the major and minor
semi-axes of the ellipse, then ε =
–
–
a 2 − b2 / a .
parabolic orbit, ε = 0, may be thought of as an infinitely elongated
ellipse, which does not close into a loop like a regular ellipse, but it
is opened at the other end (a → ∞ ).
hyperbolic orbit, for which both ε and “a” are negative numbers.
All three options are shown in Fig. 9. The velocity vector, defined by Eq.
(16), remains tangent to the orbit at all times and it can be evaluated by
application of the rules of vector algebra, namely

v (θ ) v=
=
(θ )
 dr 
2  dθ 
  +r 

dt
 
 dt 
2
2
(27)
in where r(t) and θ(t) are defined by Eqs. (21), while the angular velocity
is given by
rv
dθ
= 0 0
dt r 2 (θ )
(28)
Omitting some algebraic details, we shall provide the final expression for
the velocity of an object traveling along the orbit described by Eq. (26),
which turns out to be a following function of time and trajectory parameters
65
1/ 2

 1
1 
v (θ )  2GM 
=
−  

 r (θ ) 2a  
(29)
This equation applies to all three types of orbits and it also works for
an arbitrary mass M of the large body generating the gravitational field
(not necessarily just Earth). The universal gravitational constant
G = 6.67422 x10-8 cm3/g.s2, while the length parameter “a” can be related
to the period of rotation T (for closed orbits). Specifically, for a circular orbit when “a” denotes the radius of the orbit, the relation reads
a3 =
T 2GM
4π 2
(30)
Let us illustrate use of formula (30) in evaluating the radius of a geosynchronous orbit (known also as a geostationary, since to an observer on
Earth an object placed on this particular orbit appears to be motionless).
For such an orbit, of course, the period T must equal 24 hours. When we
substitute this T (truly, the exact number for T is 23 hrs 56 min 4 sec),
mass of Earth M E and G into Eq. (30), we obtain
a =
3
( 23.9344 hrs )
2
( 6.67422 ×10
-8
cm3g -1s -2 )( 5.97224 ×1027 g )
4π 2
= 74.9597 ×1027 cm3
Hence
a geo = 4.21641x109 cm, or a geo = 42,164.1 km.
If the mean equatorial radius of Earth R eq = 6378.1 km is subtracted from
the number given above, we arrive at the altitude of the geosynchronous
satellite, namely, h gs = 35,785.9 km.
To conclude this section we shall calculate
(1)
orbital velocity for a geosynchronous orbit, and
(2)
escape velocity for Earth.
(1)For a circular orbit we substitute r = a geo = const. into Eq. (29), reducing it to
1/ 2
vorb =  GM 
 a 
(31)
Since the radius “a” of the geosynchronous orbit is known (see the calculations above), we arrive at the orbital velocity vorb = 3.07 km/s.
(2)For the parabolic orbit, which corresponds to a trajectory without an
end, the parameter (1/a) equals zero, and then Eq. (29) predicts the socalled escape velocity
66
1/ 2
 2GM E 
 2GM 
=
vesc =



 r  Earth  RE 
1/2
(32)
Substituting the mass of Earth M E = 5.97224x1024 kg for M and the radius
R E = 6370 km for r in Eq. (29), yields
v esc = 11.1870 km/s
(33)
Figure 10. Example of a perturbation of an elliptical orbit of a planetoid (or
a spaceship) circling the Sun caused by an interaction with the Earth gravitational field. This so-called “close encounter” event visibly alters the
original orbit, as seen by the segment ABCD of the trajectory depicted in
the figure. A “three body problem” has to be considered between points A
and D, where an exchange of the mutual forces between three objects
(Sun, Earth and the planetoid) must be accounted for. The closed form solution to such a problem is not available. In the Cassini mission this situation occurs each time the spaceship enters the “sphere of influence” of
another planet on its path, such as Earth, Venus and Jupiter, which are
used to accomplish a gravitational assist maneuver.
This velocity is substantially higher that the orbital velocity calculated for
the geosynchronous orbit. The ratio of these two velocities equals about
3.64, and it represents one of the universal constants that rule our Universe (for more information about such numbers the reader is referred to
“From Pyramids and Fibonacci Sequence to the Laws of Chaos” by Michael
Wnuk and Carl Swopes, publ. by AKAPIT, and available from Panslavia Institute of Milwuakee, WI). If the orbital velocity for the geostationary object is compared with the escape velocity calculated for this height (4.3482
67
km/s), then it turns out that v orb is just about 71% of the escape velocity
pertinent to the geostationary orbit.
When more than two bodies interact via gravitational pull (see Fig. 10),
the mathematical problems become so complex that they cannot be solved
in a closed form. The governing partial differential equations in such
a case must be integrated numerically, and high-speed computers, which
are available to us today, accomplish this feat. Fig. 10 illustrates an example of the so-called “three body problem”, involving a close encounter of
a planetoid (or a spacecraft) orbiting around Sun, with our own planet.
The case considered here resembles a “gravitational assist” maneuver of
some NASA designed spacecrafts. Cursory inspection of Fig. 10 reveals
that the trajectory resulting under the conditions of such close encounter
event substantially differs from the regular elliptical orbit predicted for the
case when only two bodies are interacting with each other. The shape of
the trajectory shown in Fig. 10 could not have been obtained without help
from a modern computer. Yet, the principles of these calculations are identical to those described by Newton in his “Principia”. Newton’s analytical
determination of the elliptic orbits of planets, which had been deduced
from observational data gathered by Johannes Kepler (1571–1630), was
a milestone in the history of natural science. One may add that the design
and intricate navigation of the Cassini spaceship is a perfect example of
the continuity of Celestial Mechanics that has began several centuries ago,
and which had laid the foundation and inspired the present day space exploration programs.
4. Scalars, vectors, quaternions, matrices and tensors
All the mathematical entities discussed so far, such as numbers and functions (which connect one set of numbers with another) are examples of
scalars, or scalar fields. For example, if one wishes to describe the distribution of temperature in one’s living room as a function of location (x, y,
z) and time t, say
F = F(x, y, z, t)
(34)
Then for every point (x, y, z) within the room and at a specified time t, the
function F determines a single property: the temperature. This single
property is an example of a scalar – or, like in the case discussed here,
where the distribution of temperature is considered – we refer to it as
a scalar field.
68
Figure 11. Cartesian coordinate system (x, y, z) with the corresponding unit
vectors (i, j, k). A vector PP’ can be represented by its components [PP’x,
PP’y, PP’z], or by this equation: PP’ = (PP’x)i + (PP’y)j + (PP’z)k. If plane (x,
y) is chosen as the plane in which Earth circles the Sun, the (x, y, z) coordinates shown here represent J2000 inertial reference frame.
Unit vector and the rotation are used to define a quaternion.
The picture gets considerably more complicated when we ask a question
about the distribution of the velocities of the dust particles in the same
room. Now, the single number is not sufficient to provide a complete information about the velocity. Velocity is a vector, and in 3D space a vector can be represented by a set of three ordered numbers, for example




v=
−1.5i + 2 j + 1.75k [ m / sec ]
(35)
or, in form of a column vector
 −1.5 
 

v =  2  m/sec
1.75 


(36)
To interpret these numbers correctly, one should envision a particle traveling in the negative x-direction (backwards) with a speed of 1.5 m/sec,
while the other two components of the vector v, 2 m/sec and 1.75 m/sec,
inform us how fast is the particle moving in the y and z directions, respectively. Only after all these three numbers are provided, we are able to
reconstruct the complete velocity vector v. Using a somewhat different
terminology, we can describe a vector by a pointed arrow, see Fig. 11,
which possesses three properties:
–
orientation (a line in space),
–
sense (direction of the arrow), and
–
magnitude (length of the arrow).
From the high school Math, the length, or the magnitude, of a vector can
be calculated from this simple formula
69

v v 
 PP    PP    PP 
' 2
x
' 2
y
' 2
z
(37)
For the example used in Eqs. (35) and (36), we obtain
v
 1.5   2   1.75 
2
2
2
 3.0516 m / sec
(38)
Now, consider a spacecraft moving through the 3D space. Its velocity, of
course, is a vector. On August 18, 1999, when the Cassini had a “close
encounter” with Earth during the gravitational assist maneuver, its velocity
was




v  18.0176i  5.32693 j  2.99920k  km / s 

v v 
18.0176    5.32693   2.9992 
2
2
2
 19.03km/s
(39)
Measured in terms of the escape velocity, which is a convenient yard-stick
for measuring velocities of cosmic nature, the velocity of Cassini given by
Eq. (39) exceeded the escape velocity pertinent to this altitude by 85%.
The general case of the motion of a spaceship can be thought of as
a superposition of
–
three translations, each occurring along one of the reference axes,
x, y, z, as shown in Fig. 11, and
–
three rotations, usually defined by the Euler angles.
Therefore, we need a total of six equations of motion, three for the translations and three for the rotations. Mathematical representation of the
translational motion is relatively simple. If we adapt the notation

x, 
y and 
z for the components of the acceleration vector
x
 
  
a   
y
 

z 
Then the Newtonian equations of motion read
   Fx
mx
   Fy
my
(40)
mz   Fz
in which m is a scalar and it denotes the total mass of the moving object,
while the set of functions ( 
x, 
y, 
z ) represents the acceleration vector, and
the force field is described by yet another vector
F 
  x
F   Fy 
F 
 z
70
(41)
To define the three rotations shown in Fig. 11, we use the traditional aviation terminology, namely
pitch – rotation about the x-axis, αx
roll – rotation about the y-axis, α y
yaw – rotation about the z-axis, α z .
Here we have assumed that the spacecraft (or, an aircraft) is moving in
the direction aligned with y, and its nose is located at the origin of the
coordinate system shown in Fig. 11. The three angles, listed above, are
known as the Euler angles, and are commonly used in the Dynamics of 3D
solid bodies. Equations that relate these rotations to the applied moments,
or better, a moment vector
M 
  x
M = My 
M 
 z
(42)
are again dictated by the Newtonian mechanics, and they read
 I11 I12 I13   αx   ∑ M x 


    
α
I
I
I
M
=


∑
y
y
21
22
23

 

 I I I   α  
 31 32 33   z   ∑ M z 
(43)
Two dots placed over the Euler angles signify the second time derivative,
or angular acceleration, which – again – is a vector
α x 
2
 d  

αy
a =
=
dt 2  
α z 
 αx 
 
 αy 
 α 
 z
(44)
Note that in “short-hand” notation Eq. (43) can be written as follows


Iα = ∑ M

Here, the quantity
(45)
I is the 3 by 3 matrix of moments of inertia, which

measures body inertial resistance to a rotation. In fact, due to the special
rules of transformations, with which I complies when transformed from
one reference frame to another, it is also a tensor.
Since the moment of inertia matrix (or, tensor) is a symmetric matrix, out
of 9 of its components only 6 are independent. To be a little more specific,
let us take a closer look at the Cassini’s tensor of moment of inertia
71
 9362 129 − 117 


=
I cassini  129 9652
52   kg ⋅ m 2 

 −117 52 3982 


(46)
These numbers resulted from the specific distribution of mass of the Cassini spacecraft (see the cover page of this report). The tensor I can be

reduced to the diagonal form
0 
 9314 0


=
I dia  0 9703 0   kg ⋅ m 2 

 0
0 3978 

(47)
where the non-zero diagonal elements designate the principal values
(largest, intermediate and smallest) of the moment of inertia for the Cassini spaceship. They were obtained by a rather tedious process of seeking
the eigenvalues for the matrix (46). The “bottom line” of this process consists in expanding the determinant
I11 − λ ,
I12 ,
I13
I 21 ,
I 22 − λ ,
I 23
I 31 ,
I 32 ,
I 33 − λ
0
=
(48)
and then solving the resulting cubic equation
λ 3 − Ω1λ 2 + Ω 2 λ − Ω3 = 0
(49)
The so-called invariants Ω 1 , Ω 2 and Ω 3 are computed from the known
components of the matrix as follows
Ω1= I11 + I 22 + I 33
Ω
=
I11 I 22 + I 22 I 33 + I 33 I11 − I122 − I 232 − I 312
2
(50)
=
Ω3 I11 I 22 I 33 + I12 I 23 I 31 − I11 I 232 − I 22 I132 − I 33 I122
For the Cassini spacecraft, using the data contained in Eq. (46),we obtain
Ω1 =−22,995 kg m 2
166, 023, 724 kg 2 m 4
Ω2 =
(51)
Ω3 =−3.5951×1011 kg 3 m 6
When these numbers are substituted into the cubic equation (49), and
then the equation is solved, we arrive at the three real roots, shown in the
diagonal of the expression (47). These diagonal components of the matrix
determine the moments of inertia about the three principle axes of the
spaceship. The fact that we did indeed find the three real roots means that
72
the matrix is “positive definite”, as expected, and that our problem of
seeking the eigenvalues and eigenvectors, is “well posed” as a mathematician would say. On a more engineering side we can judge how the spaceship is “balanced” with respect to its center of mass; the first two numbers
shown on the diagonal in matrix (47) are quite similar (which is good),
while the third one indicates moment of inertia with respect to the third
major axis that almost coincides with the ship vertical axis, cf. the figure
on the cover page. This means that the ship will easily roll, while it is more
resistant to pitch and yaw.
As it can be appreciated from these considerations, using the matrix
the Euler angles

α
I and

gets rather complex and tedious. It is possible, though,
to replace this formalism by another representation involving just one vec
tor of unit length, η , and just one rotation marked by θ in Fig. 11. To
represent a vector in 3D we need a set of three numbers, and to represent
an angle we need one scalar. Therefore, total of four pieces of information
are necessary to describe the attitude (orientation) of the spaceship on its
orbit. In this way the matrix algebra is avoided altogether. What we get
instead is an ordered set of four numbers, the so-called quaternion, which

incorporates the orientation in space (vector η ) and the angle of rotation,
θ. Cassini’s attitude quaternion on its present orbit reads



q̂= -0.5421 -0.3389 i -0.1594j -0.0146k
A “hat” is used to distinguish a quaternion from a vector
(52)

v and/or a tensor
I . The first number shown on the right side of Eq. (52) provides an infor
mation about the rotation θ (yes, the ship rotates as it travels along its
orbit), while the other three numbers define the orientation of the unit


vector η . They are related to the directional cosines of the η -vector, η 1 ,
η 2 , η 3 and η 4 as follows:
q 1 = cos(θ/2)
(53)
q 2 = η 1 sin(θ/2)
q 3 = η 2 sin(θ/2)
q 4 = η 3 sin(θ/2)
When all these quantities are put together, we recover a quaternion
q= [ q1 , q 2 , q 3 , q 4 ]
(54)
For any unit vector, of course, the sum η12 + η22 + η32 equals 1.
And the directional cosines are cosines of the angles formed between the
vector η and the corresponding coordinate, η i = cos(x i ,η). Here, for simplicity the index “i” has been applied to number the axes, so that instead
x, y and z, we have x 1 , x 2 and x 3 . The most common choice of the reference frame (x 1 , x 2 , x 3 ) is the J-2000 inertial frame attached to Earth in
such a way that the x 3 axis (or, z axis in Fig. 11) is slanted at 23o from the
polar axis of the planet. The tilt is necessary to align the reference coordi-
73
nates with the ecliptic plane to which our planet is confined in its motion
around Sun. As the z-axis of the J-2000 frame is perpendicular to the ecliptic plane, while the polar axis of Earth is not, one needs to account for
the 23o angle between the two axes.
We conclude this section with a brief overview of the characteristic parameters (all scalars, of course) describing three planets: Venus, Earth and
Saturn. For details see Table 3.
PLANET
Mass Relative to
Mass of Earth (a)
Average Distance
From Sun in AU (b)
Eccentricity of
the Orbit
Period (in Earth
years) of Motion
Around Sun
Average Orbital
Speed (KMS –1)
Escape Velocity
From the Planet
(KMS –1)
Period of Rotation
About Polar Axis
(hrs)
Table 3. Characteristic Parameters of Three Planets
Venus
0.816
0.723
0.007
0.615
35.0
10.4
5.832
Earth
1
1
0.017
1
29.8
11.2
24
Saturn
95.2
9.54
0.056
29.46
9.6
36.2
10.3
a) Mass of Earth M E = 5.97224 x 10
COMME
NTS
24
(c)
kg
b) AU = Astronomical Unit = Average Distance of Earth from Sun (1AU) =
1.496 x 108 km
c) Venus has so-called “negative” rotation. This means that when the planet is
viewed along the polar axis from the North, it rotates clockwise.
Suggested Reading
1. John A. Wood, 1979, “The Solar System”, publ. by Prentice-Hall, New Jersey.
2. Anthony Bedford and Wallace Fowler, 1995, “Engineering Mechanics – Dynamics”, publ. by Addison-Wesley, USA.
3. David A. Vallado and Wayne D. McClain, 1997, “Fundamentals of Astrodynamics and Applications”, in Space Technology Series, publ. by McGrawHill, USA.
4. Michael P. Wnuk and Carl Swopes, 1999, “From Pyramids and Fibonacci
Sequence to the Laws of Chaos”, publ. by Akapit Publishers, Krakow, Poland.
5. Levin Santos, 2000, “Weighing the Earth. Physicists Close in on Newton’s
Big G” in the “Sciences”, July/August 2000, publ. by New York Academy of
Sciences, p.11.
74
MECHANICS OF TIME DEPENDENT
FRACTURE
Michael P. Wnuk1
College of Engineering and Applied Science,
University of Wisconsin-Milwaukee, Email: [email protected]
ABSTRACT
Effects of two parameters on enhancement of the time-dependent fracture manifested by a slow stable crack propagation that precedes catastrophic failure in ductile materials have been studied. One of these parameters is related to the material ductility (ρ) and the other describes the geometry (roughness) of crack surface
and is measured by the degree of fractality represented by the fractal exponent α,
or – equivalently – by the Hausdorff fractal dimension D for a self-similar crack.
These studies of early stages of ductile fracture are preceded by a brief summary
of modeling the phenomenon of delayed fracture in polymeric materials, sometimes referred to as “creep rupture”. Despite different physical mechanisms involved in the preliminary stable crack extension and despite different mathematical
representations, a remarkable similarity of the end results pertaining to the two
phenomena of slow crack growth (SCG) that occur either in viscoelastic or in ductile media has been demonstrated.
1. Crack motion in a viscoelastic medium
In late sixties and early seventies of the past century a number of physical
models and mathematical theories have been developed to provide a better insight and a quantitative description of the early stages of fracture in
polymeric materials. In particular two phases of fracture initiation and
subsequent growth have been considered: (1) the incubation phase during
which the displacements of the crack surfaces are subject to creep process
but the crack remains dormant; and (2) slow propagation of a crack embedded in a viscoelastic medium. According to the linear theory of viscoelastic solids, the material response to the deformation process obeys the
following constitutive relations
t
∫ G (t − τ )
sij=
(t , x)
1
0−
t
s=
(t , x)
∫ G (t − τ )
2
0−
∂eij (τ , x)
∂τ
dτ
(1.1)
∂e(τ , x)
dτ
∂τ
Here s ij is the deviatoric part of the stress tensor, s denotes the spherical
stress tensor, while G 1 (t) and G 2 (t) are time dependent relaxation moduli
for shear and dilatation, respectively. The inverse relations read
75
t
∫ J (t − τ )
eij =
(t , x)
1
0−
∂sij (τ , x)
∂τ
dτ
(1.2)
∂s (τ , x)
e=
(t , x) ∫ J 2 (t − τ )
dτ
∂τ
0−
t
Symbols e ij and e are used to denote the deviatoric and spherical strain
tensors and J 1 (t) and J 2 (t) are the two creep compliance functions. For
a uniaxial state of stress these last two equations reduce to a simple form
t
∫ J (t − τ )
=
ε (t )
0−
∂σ (τ )
dτ
∂τ
(1.3)
The relaxation moduli G 1 (t), G 2 (t) and the creep compliance functions
J 1 (t) and J 2 (t) satisfy the following integral equations
t
t
∫ G (t − τ ) J (τ )dτ =
1
1
0−
t
∫ G (t − τ ) J
2
2
(1.4)
(τ )dτ =
t
0−
For a uniaxial state of stress these equations reduce to a single relation
between the relaxation modulus E rel (t) and the creep compliance function
J(t)
t
∫E
rel
t
(t − τ ) J (τ )dτ =
(1.4a)
0−
Atomistic model of delayed fracture was considered by Zhurkov (1965),
but this molecular theory had no great impact on the further development
of the theories based in the Continuum Mechanics approaches. Inspired by
Max Williams W. G. Knauss of Caltech in his doctoral thesis considered
time dependent fracture of viscoelastic materials, Knauss (1965). Similar
research was done by Willis (1967) followed by simultaneous researches of
Williams (1967, 1968, and 1969), Wnuk and Knauss (1970), Field (1971),
Wnuk (1968, 1969, 1971, and 1972), and also by Knauss and Dietmann
(1970), Mueller and Knauss (1971a, 1971b), Graham (1968, 1969), Kostrov and Nikitin (1970), Mueller (1971), Knauss (1973) and Schapery
(1973).
What follows in this section is an attempt to present a brief summary of
the essential results, which have had a permanent impact on the development of the mechanics of time dependent fracture. After this review is
completed we shall indicate an interesting analogy of delayed fracture in
polymers (intricately related to the ability to creep) with the “slow crack
growth” (SCG) occurring in ductile solids due to the redistribution of
strains within the yielded zone preceding the front of a propagating crack.
Two stages of delayed fracture in viscoelastic media, incubation and
agation, are described respectively by two governing equations: (1) WnukKnauss equation and (2) Mueller-Knauss-Schapery equation. The duration
76
of the incubation stage can be predicted from the Wnuk-Knauss equation
2
J (t1 )  K G 
Ψ (t1 )=
= 

J (0)  K 0  a=
(1.5)
a0= const
o
Mueller-Knauss-Schapery equation relates the rate of crack growth a to
the applied constant load σ 0 and the material properties such as the unit
step growth Δ, usually identified with the process zone size, and the Griffith stress σ = 2 Eγ
G
π a0
, namely
o
 ∆  J (∆ / a)  K 2
Ψ  o=
=  G

 
J
(0)
 K0 
a
(1.6)
For a constant crack length equal the length of the initial crack a 0 , the
right hand side in (1.5) reduces to the square of the ratio of the Griffith
stress to the applied stress
σ 
n= G 
 σ0 
2
(1.7)
This quantity is sometimes referred to as “crack length quotient” – it determines how many times the actual crack is smaller than the critical Griffith crack. Therefore, the larger is the number “n”, the further away is the
initial defect from the critical point of unstable propagation predicted for
a Griffith crack embedded in a brittle solid. For large “n” the crack is too
short to initiate the delayed fracture process, see expression (1.13a) for
the definition of the n max . Beyond n max growth of the crack cannot take
place. For n > n max one can assume that theses are stable cracks, which –
according to the theory presented here – will never propagate. These are
so-called “dormant cracks” that belong to a “no-growth” domain, see Appendix.
When crack length “a” is not constant, but it can vary with time a = a(t),
then the right side in (1.6) reads
2
 σ G  a0 n
=


 σ0  a x
(1.8)
Here x denotes the non-dimensional crack length, x=a/a 0 . It is noteworthy
o
that the physical meaning of the argument ∆ / a appearing in (1.6) is the
time interval needed for the tip of a moving crack to traverse the process
zone adjacent to the crack tip, say
o
δt = ∆ / a
(1.9)
The location of the process zone with respect to the cohesive zone which
precedes a propagating crack is shown in Fig. 1.
77
Fig. 1. Structured cohesive zone crack model of Wnuk (1972, 1974). Note that of
the two length parameters Δ and R the latter is time dependent analogous to
length a, which denotes the length of the moving crack. Process zone size Δ is the
material property and it remains constant during the crack growth process. Ratio
R/ Δ serves as a measure of material ductility; for R/ Δ>>1 material is ductile,
while for R/ Δ -> 1, material is brittle.
To illustrate applications of the equations (1.5) and (1.7) we shall use the
constitutive equations valid for the standard linear solid, see Fig. 2. With
β 1 denoting the ratio of the moduli E 1 /E 2 the creep compliance function for
this solid is given as
J (t ) =
1
{1 + β1 [1 − exp(−t / τ 2 )]}
E1
E2
(1.10)
τ2
E1
Fig. 2. Schematic diagram of the standard linear solid model.
Therefore, the nondimensional creep compliance function
reads
Ψ (t ) =1 + β1 [1 − exp(−t / τ 2 ) ]
Ψ (t)=J(t)/J(0)
(1.11)
Substituting this expression into (1.5) one obtains
1 + β1 [1 − exp(−t1 / τ 2 ) ] =
n
78
(1.12)
Solving for t 1 one obtains the following prediction for the incubation time
valid for a material represented by standard linear solid

β1 
t1 = τ 2 ln 

 1 + β1 − n 
(1.13)
Inspection of (1.13) reveals that the quotient “n” should not exceed a certain limiting level
nmax = 1 + β1
(1.13a)
Physical interpretation of this relation can be stated as follows: for short
cracks, when n>n max , there is no danger of initiating the delayed fracture
process. These subcritical cracks are permanently dormant and they do
not propagate.
Fig. 3a illustrates the relationship between the incubation time and the
loading parameter given either as n or s(= 1/ n = σ 0 /σ G ). Fig. 3b shows
an analogous relation between the time used in the process of crack propagation and the loading parameter s. Note that the incubation time is expressed in units of the relaxation time τ 2 , while the time measured during
the crack propagation phase of the delayed fracture is expressed in units
of (τ 2 /δ), in where the constant δ contains the initial crack length a 0 and
the characteristic material length Δ, cf. (1.16). When the variable s is used
on the vertical axis and the pertinent function is plotted against the logarithm of time, then it is seen that a substantial portion of the curve appears as a straight line. This confirms the experimental results of Knauss
and Dietmann (1969) used also by Schapery (1973) and Mohanty (1972).
NONDIMENSIONAL LOAD, s=σo/σG
0.7
0.6
0.5
0.4
ß1 =10
ß1 = 100
0.3
0. 01
0.1
1
10
LOGARITHM (TIME/τ2)
Fig. 3a. Logarithm of the incubation time in units of τ 2 shown as a function of the
loading parameter s for two different values of the material constant β 1 = E 1 /E 2 .
79
NONDIMENSIONAL LOAD, s=σo/σG
0.7
0.6
0.5
0.4
0.3
0.01
β1 =100
β1 =10
0.1
1
10
LOGARITHM (CRITICAL TIME/(τ2/δ))
Fig. 3b. Logarithm of the time-to-failure used during the crack propagation phase,
in units of τ 2 , shown as a function of the loading parameter s for two different values of the material constant β 1 = E 1 /E 2 .
To describe motion of a crack embedded in viscoelastic solid represented
by the standard linear model one needs to insert (1.10) into the governing
equation (1.6). The equation of motion reads then
n
1 + β1 [1 − exp(−δ t / τ 2 ) ] =
x
(1.14)
o
Solving it for the time interval δt/τ 2 (= ∆ / a τ 2 ) yields




β1
= ln 

o
n
 1 + β1 − 
τ2 a
x

∆
(1.15)
It is seen from (1.15) that for the motion to exist, the quotient n should
not exceed the maximum value defined by (1.13a). For n > n max the
cracks are too small to propagate.
If nondimensional notation for the length and time variables is introduced
δ = ∆ / a0
θ = t /τ2
(1.16)
the left hand side of (1.15) can be reduced as follows
δt
=
τ2
80
∆
a0
∆
∆
=
=
o
d ( xa0 )
dx
τ2 a
τ2
dθ
d (θτ 2 )
(1.17)
When this is inserted into (1.15) and with δ = Δ/a 0 , the following differential equation results
 

 

β1
dx
= δ ln 

dθ
  1 + β1 − n  
 
x  
−1
(1.18)
or, after separation of variables




β1
(δ )dθ = ln 
 dx
 1 + β1 − n 
x

(1.19)
Motion begins at the first critical time t 1 , which designates the end of the
incubation period. Therefore, the lower limit for the integral applied to the
left hand side of (1.19) should be θ 1 = t 1 /τ 2 , while the upper limit is the
current nondimensional time θ = t/τ 2 . The corresponding upper limit to the
integral on the right hand side of (1.19) is the current crack length x =
a/a 0 , while the lower limit is one. Upon integration one obtains


x


β1
1
dθ =   ∫ ln 
 dz
∫
n
 δ  1  1+ β − 
t1 /τ 2
1

z
t /τ 2
(1.20)
The resulting expression relates the crack length x to time t, namely


x


β1
τ2 
t − t1 =
 dz
  ∫ ln 
 δ  1  1+ β − n 
1
z

(1.21)
If the closed form solution for the integral in (1.21) is used, then this formula can be cast in the following final form


 (1 + β1 ) x − n 
 1 + β1 − n  
xβ1
n
 τ  
t=
t1 +  2   x ln 
ln 
  (1.22)
+
 + ln 
+
−
+
+
−
(1
β
)
x
n
1
β
1
β
n
β
 δ  





 
1
1
1
1
This equation has been used in constructing the graphs shown in Fig. 4. At
β 1 = 10 three values of n have been used (4.00, 6.25 and 8.16, which corresponds to the following values of s: 0.5, 0.4 and 0.35). It can be observed that at x approaching n the phase of the slow crack propagation is
transformed into unrestrained crack extension tantamount to the catastrophic fracture. The point in time, at which this transition occurs, can be
easily seen on the horizontal axis of Fig. 4. This point of transition into unstable propagation can also be predicted from (1.22); substituting n for x
we obtain the time to fracture
 β1n 
1 + β1 − n  
 τ   n
t2  2  
ln 
=
 + ln 

 δ  1 + β1 1 + β1 − n 
 β1  
(1.23)
81
If the incubation time t 1 given by (1.13) is now added to (1.23), one obtains the total life time of the component, namely

 β1n 
1 + β1 − n  
β1   τ 2   n
Tcr = t1 + t2 =τ 2 ln 
ln 
+ 
 + ln 

 1 + β1 − n   δ  1 + β1 1 + β1 − n 
 β1  
(1.24)
Summarizing the results of this section we can state that the delayed fracture in a viscoelastic solid can be mathematically represented by four expressions:
–
time of incubation t 1 given by (1.13) for standard linear model,
–
equation of motion given by (1.22) for the same material model
and defining x as a function of time, x = x(t),
–
time to fracture t 2 due to crack propagation given by (1.23),
–
life time T cr equal to the sum t 1 + t 2 , as given by (1.24). It is
noted that while the first term in the expression (1.24) involves
the relaxation time, material constant β 1 and the quotient n, the
second term in (1.24) contains also the internal structural constant δ. It is also noted that for the quotient n approaching one,
both terms in (1.24) are zero, while for n exceeding n max , the
expression looses the physical sense (since in that case there is
no propagation). With the constant δ being on the order of
magnitude varying within the range 10-3 to 10-6 the second
term in (1.24) is substantially greater than the first term which
represents the incubation time, see also Appendix.
Fig. 4. Slow crack propagation occurring in a linear viscoelastic solid represented
by the standard linear model depicted in Fig. 2 at β 1 = 10. Crack length is shown
as a function of time; points marked on the negative time axis designate the incubation times corresponding to the given level of the applied constant load n and
expressed in units of τ 2 . The time interval between the specific point t1 and the
origin of the coordinates provides the duration of the incubation period. Crack
propagation begins at t = 0. Symbol t 2 denotes time-to-failure, which is the time
used during the quasi-static phase of crack extension and it is expressed in units of
(τ 2 /δ). Constant δ is related to the characteristic material length, the so-called “unit
growth step” Δ.
82
For β 1 = 10 and three different levels of n, the resulting functional relationships between the crack length x and time t are shown in Fig. 4 along
with the values of the incubation times, expressed in units of (τ 2 ), and the
times-to-failure expressed in units of (τ 2 /δ). A numerical example is given
in the Appendix.
Example described here, involving the standard linear solid, serves as
an illustration of the mathematical procedures necessary in predicting the
delayed fracture in polymeric materials. Knauss and Dietmann (1969) and
Schapery (1973) have shown how the real viscoelastic materials, for which
the relaxation modulus G(t) and the creep compliance function J(t) are
measured (or calculated from equation (1.4)) and then used in the governing equations of motion discussed above can provide a good approximation of the experimental data.
2.
Quasi-static stable crack propagation in ductile solids
Crack embedded in a ductile material will tend to propagate well below the
threshold level indicated by the ASTM standards. This phenomenon of slow
crack growth (SCG) is sometimes referred to as “subcritical” or “quasistatic” crack propagation and it is caused by the redistribution of elastoplastic strains induced at the front of the propagating crack. The higher is
the ductility of the material, the more pronounced is the preliminary crack
extension associated with the early stages of fracture. For brittle solids
this effect vanishes.
Ductility of the material is defined as the ratio of two characteristic strains,
namely
ρ=
εf
εf
= 1 + pl
εY
εY
(2.1)
Here εf denotes strain at fracture, and it can be expressed as the sum of
the yield strain ε Y and the plastic component of the strain at fracture εf pl .
We will refer to the material property defined by (2.1) as ductility index
and we shall relate it to the parameters inherent in the structured cohesive zone crack model, cf. Wnuk (1972, 1974) – see also Fig. 1. According
to Wnuk and Mura (1981, 1983) the relation is as follows
ρ=
Rini
∆
(2.2)
Here the symbol R ini denotes the length of the cohesive zone at the onset
of crack growth, while Δ is the process zone size or the so-called “unit
growth step” for a propagating crack. In order to mathematically describe
motion of a quasi-static crack one needs to know the distribution of the
opening displacement within the cohesive zone of the crack shown in Fig.
1. When the cohesive zone is much smaller than the crack length (this is
83
the so-called Barenblatt’s condition) according to Rice (1968) and Wnuk
(1974) this distribution is established as follows
u=
y ( x1 , R )
4σ Y
π E1

x1  R + R − x1  

 R( R − x1 ) − ln 
2  R − R − x1  

(2.3)
Here x 1 denotes the distance measured from the physical crack tip, E 1 is
the Young modulus E for the case of plane stress, while for the plane
strain it is E(1-ν2)-1 where ν is the Poisson ratio. Symbol σ Y denotes the
yield stress present within the end zone. For a moving crack both x 1 and R
are certain functions of time – or, equivalently – of the crack length a,
which can be used here as a time-like variable. In agreement with Wnuk’s
“final stretch criterion”, cf. Wnuk (1972, 1974), two adjacent states of the
time-dependent structured cohesive zone should be examined simultaneously, as shown in Fig. 5. At the instant t (state 2 in Fig. 5) the opening
displacement u y (x 1 (t),R(t)) measured at the control point P, say u 2 (P),
equals
u2 ( P )
=
4σ 0
4σ 0 
dR 
R ]x = 0
∆
[=
[ R ]x1 = ∆ +
1
da 
π E1
π E1 
(2.4)
Expansion of the variable R(x 1 ) into a Taylor series is justified, since both
states considered are in close proximity. For simplicity the entity
[ R ]x = ∆
1
shall be referred to as R(Δ). Note that at the preceding instant “t-δt” then
(state 1 in Fig. 5) the vertical displacement u y within the cohesive zone,
measured at the control point P, located at x 1 = Δ for state 1, equals
u1 ( P)
=
4σ 0 
∆  R(∆) + R(∆) − ∆  

 R(∆)( R(∆) − ∆) − ln 
2  R(∆) − R(∆) − ∆  
π E1 
(2.5)
Fig. 5. Distribution of the COD within the cohesive zone corresponding to two subsequent states represented by instants “t” and “t-δt” in the course of quasi-static
crack extension as required in Wnuk's criterion of delta COD; [v 2 (t) – v 1 (t- δt)] P =
final stretch.
84
According to Wnuk’s “delta COD” or “final stretch” criterion for crack motion to occur it is necessary that the difference between (2.4) and (2.5) is
maintained constant and equal to the material parameter
∧
δ/ 2,
where
∧
δ
is the final stretch regarded invariant during the crack growth process.
Note that a similar requirement is postulated for the size of the process
zone or unit growth step, Δ = const. Therefore, the final stretch criterion
reads
∧
u2 ( P) − u1 ( P) =
δ/2
(2.6)
Substituting (2.4) and (2.5) into the criterion of subcritical motion (2.6)
and naming R(Δ) by R, one obtains the following differential equation
∧
∆  R + R − ∆  δ  π E1 
dR
− R( R − ∆) + ln 
R+∆
= 

2  R − R − ∆  2  4σ Y 
da
We note that while both
∧
δ
(2.7)
and Δ are constant, the entity R is a certain
unknown function of the crack length a. Using the nondimensional length
of the cohesive zone, Y and the nondimensional crack length X
Y=
R
Rini
X=
a
Rini
(2.8)
and denoting the group of material constants on the right hand side of
(2.7) by M and referring to it in the sequel as “tearing modulus”
∧
δ πE 
M =  1
2  4σ Y 
(2.9)
we rewrite the governing differential equation (2.7) in this form
dY
1  ρY + ρY − 1 
= M − ρY + ρY ( ρY − 1) − ln 

dX
2  ρY − ρY − 1 
(2.10)
This equation can be further reduced if it is assumed that we focus the attention on the ductile material behavior, when R>>Δ, and therefore consider the case when the ductility index ρ substantially exceeds one. Physically it means that the process zone Δ is much smaller than the length of
the cohesive zone. With such an assumption and some algebraic manipulations involving expansion of the pertinent functions into power series one
may reduce the right hand side of (2.10) to the following simple form, cf.
Wnuk (1972, 1974) and Rice et al. (1978 and 1980)
dY
1 1
= M − − ln(4 ρY )
dX
2 2
(2.11)
Slow crack growth is possible only if the initial slope of (2.11) is positive,
i.e.,
85
 dY 
 dX  ≥ 0
Y =1
(2.12)
This condition imposes a certain restriction on the tearing modulus M. For
motion to take place M must be greater than a certain minimum tearing
modulus, i.e.,
M min=
1 1
+ ln(4 ρ )
2 2
(2.13)
To illustrate applications of the governing equation (2.11) we shall assume
in what follows that the tearing modulus M(ρ) is 10% higher than the minimum modulus defined by (2.13)
1 1

M=
( ρ ) 1.1  + ln(4 ρ ) 
2 2

(2.14)
Now we focus attention on the differential equation (2.11) amended by the
condition (2.14), namely
dY
1 1
= M ( ρ ) − − ln(4 ρY )
dX
2 2
(2.15)
It is noteworthy that according to the cohesive crack model the length R
differs only by a multiplicative constant from Rice’s J-integral and from the
Wells (1963) opening displacement”, COD. Denoting the COD by δ tip , we
recall the following well-known relations valid for a cohesive crack model
under the restriction of small scale yielding when the Barenblatt condition
applies
J = σ Y δ tip
 8σ Y 
R
 π E1 
δ tip = 
(2.16)
 8σ 
J =
R
 π E1 
2
Y
β=
(a)
σ (a) 2 2 R(a) 2 2Y ( X )
=
=
σY
π
π
a
X
The last equation in (2.16) represents the Dugdale relation between the
length of the cohesive zone R and the applied load σ valid for
a propagating crack for which both σ and R are certain functions of the
crack length, while R is subjected to the Barenblatt condition R<<a. When
physical interpretation is applied to the equations listed in (2.16), one
comes to a conclusion that the material resistance J R (a) due to continuing
crack growth can be readily represented by the resistance curve R =R(a),
or Y = Y(X). Denoting the ratio σ/σ Y by β, we rewrite the last of the equations (2.16) as follows
86
β (X ) =
2
π
2Y ( X )
X
(2.17)
Of course, β defined by (2.17), is a function of X. Let us now denote the
right hand side of the governing differential equation of a moving crack by
F(Y,ρ). Equation (2.15) thus reads
dY
= F (Y , ρ )
dX
(2.18)
Solution of (2.18) is readily obtained by the separation of variables followed by the integration, namely
Y
) X0 + ∫
X (Y=
1
1
dz
F ( z, ρ )
(2.19)
Examples of the material resistance curves Y = Y(X), or J R = J R (a), that
result from (2.19) are shown in Figure 6. It is seen that the level of material ductility ρ has a substantial influence on the slope and shape of such
material resistance curves.
MATERIAL RESISTANCE TO CRACK, Y=R/Rini
1.8
1.6
ρ =80
ρ =40
1.4
ρ =20
1.2
1
11
10
12
13
14
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
Fig. 6. Material resistance curves obtained for three different levels of material
ductility ρ = 20, 40 and 80 and for the initial crack length a 0 = 10R ini . Points of
terminal instability for each case are marked with little circles. Compared to
a brittle solid, for which ductility index approaches one, the following increases in
the effective material toughness at the transition to catastrophic fracture are observed: 36.6% at ρ = 20, 45.2% at ρ = 40 and 54.2% at ρ = 80.
Figure 7 shows the graphs illustrating dependence of the loading parameter β on the current crack length at various values of the ductility index ρ.
87
Equation (2.17) has been used to construct these curves. At
a certain value of X each such “beta-curve” attains a maximum. When the
slope dβ/dX approaches zero, the stable crack growth can no longer be
sustained. Effects of the specimen geometry and loading configuration on
the instabilities in fracture governed by equations (2.18) and (2.19) were
studied by Rouzbehani and Wnuk (2002). Some other aspects of the structured cohesive crack model and Wnuk’s criterion for subcritical crack
growth were described in Wnuk (2003a, 2003b, 2003c).
LOADING PARAMETER, β=σ/σY
0.32
ρ =80
0.31
ρ =40
ρ =20
0.3
0.29
0.28
10
11
12
13
14
NONDIMENTIONAL CRACK LENGTH, X=a/Rini
Fig. 7. Nondimensional loading parameter β (=σ/σ Y ) shown as a function of the
current crack length X = a/R ini . During the quasi-static crack extension the applied
load increases with an increasing crack length up to the point of maximum on the
beta-curve. At this point the slow crack growth process ends and the transition to
unstable (catastrophic) crack propagation takes place. Thus, the curves shown in
the figure loose their physical meaning beyond the points of maxima. Observed
increases in the loading parameter β, compared to the case of ideally brittle solid,
are as follows: 4.4% for ρ = 20, 8.3% for ρ = 40 and 10.4% for ρ = 80.
Quasi-static crack extension is viewed as a sequence of the local instability
states. Attainment of the terminal instability state, which is tantamount to
the catastrophic fracture, is seen as the termination of the slow crack
growth process. There are several techniques to establish the exact location (load and crack length) of the terminal instability state. Perhaps the
simplest approach is to seek the maximum on the beta-curve.
To do just that let us rewrite (2.17) as follows
 4  2Y
β2 = 2 
π  X
88
(2.20)
Differentiating both sides with respect to Y one gets
 dX
X −
8
 dY
2β d β = 2
π
X2

Y
 dY
(2.21)
Hence
dβ
4  1   (dX / dY )Y 
β
=
  1−

dY π 2  X  
X
(2.22)
In order to convert this expression to dβ/dX one needs to multiply it by
dY/dX defined by (2.18), which yields
dβ
4 1 
Y
F (Y , ρ ) − 
=
2

dX π β X 
X
(2.23)
For convenience we shall refer to the quantity proportional to the derivative dβ/dX as the “stability index” S = S(X), namely
dβ
dX
4
Y
S ( X ) F (Y , ρ ) −
=
X
S(X ) =
π2
Xβ
(2.24)
Examples of the plots S vs. X are given in Figure 8. As can be readily seen
all curves intersect the axis S = 0, and it is easy to read (or evaluate numerically) those zero points present in the stability indices diagrams. The
results X max , Y max and β max provide the coordinates characterizing the terminal instability states. It should be noted that the first term in the expression for the stability index S in (2.24) is proportional to the rate at
which energy is absorbed by the ductile material, while the second term is
proportional to the rate at which energy is supplied by the external force.
Both terms can be shown to be related to the second derivatives of the
potential energy of a solid weakened by a crack and subjected to certain
kind of external loading configuration corresponding to either “fixed grips”
or “constant load” boundary condition, cf. the Appendix.
In order to demonstrate the crack propagation process the diagrams
shown in Fig. 7 have been re-plotted in the way shown in Fig. 9. Here the
vertical axis represents the current crack length, while the horizontal axis
shows a nondimensional variable proportional to time. To make these
graphs as simple as possible a constant rate of load increase has been assumed. The graphs shown in this figure are remarkably similar to the
graphs shown in Fig. 4 obtained for a crack propagating through a viscoelastic medium.
Despite very different physical interpretation of the mechanisms that make
slow crack growth possible in the two considered cases, viscoelastic and
ductile media, the end results are strikingly similar.
89
STABILITY INDEX, S
Fig. 8. Stability index S shown as a function of the current length X of the propagating crack. It is noted that the function S passes through zero at the values of
length X exactly coinciding with the location of the maxima observed on the betacurves. The predicted increases in the crack length occurring due to the preliminary slow crack growth are as follows: 20.8% for ρ = 20, 23.7% for ρ = 40, and
26.5% for ρ = 80.
NONDIMENSIONAL TIME
Fig. 9. Crack length X during the quasi-static crack growth process shown as
a function of the nondimensional time. In order to construct these graphs a constant rate of load increase was assumed. At the points where the slopes of these
curves approach infinity the slow crack extension undergoes a transition into unstable (catastrophic) crack propagation. Note that this transition occurs at the values
of X corresponding to the maxima on the beta-curves shown in Fig. 7, or – equivalently – the zeros of the S-functions shown in Fig. 8.
90
3. Effect of crack surface roughness on the extent of the quasistatic crack growth. Fractal fracture mechanics
For almost all materials it is necessary to account for the roughness of the
crack surfaces. Mathematically this can be achieved by application of the
fractal model of a crack, cf. Wnuk and Yavari (2003, 2005, 2008 and
2009) and Khezrzadeh et al. (2011). The degree of fractality – proportional to the degree of roughness of the crack surfaces – is suitably measured
by the fractal exponent α, which appears in the expression for the near-tip
stress field associated with a fractal crack, namely
σ ij  r −α
(3.1)
The exponent α is related to the Hausdorff measure D of the fractal used
to represent a self-similar crack
D = 2(1- α)
(3.2)
Variation of the fractal dimension D from 1 (smooth crack) to 2 (twodimensional void) corresponds to the variation of the exponent α from ½
to zero. Therefore, for α = ½ expression (3.1) yields the relation wellknown in the Linear Elastic Fracture Mechanics (LEFM), while for the other
extreme of α approaching zero, the singularity in (3.1) disappears. Wnuk
and Yavari (2003) model of an a crack embedded in the stress field due to
a fractal geometry of the crack applies to the range of α close to 0.5 – corresponding to the range of the fractal dimension D close to 1.
In what follows we shall study the effect of the degree of fractality (measured either by α or by D) on the quasi-static crack extension, which precedes catastrophic fracture. We shall apply the formula for the opening
displacement within the cohesive zone associated with a structured cohesive crack model of Wnuk and extended to the fractal geometry, namely
u y ( x1 , R)
=
f
f

x  R + R − x1  
4σ Y

κ (α )  R f ( R f − x1 ) − 1 ln 
π E1
2  R f − R f − x1  



(3.3)
where the cohesive zone length Rf associated with a fractal crack is related
to R for the smooth crack by this expression, cf. Khezrzadeh et al. (2011)
R f = N (α , X , Y ) R
δ tipf = κ (α )δ tip
(3.4)
1
N (α , X , Y ) = N1 (α ) β ( X ) α
−2
1

α
1

−2
αΓ(α ) 
 ≈ −0.829α 3 + 1.847α 2 − 1.805α + 1.544
=
N1 (α ) 4π 2α 
1
Γ +α 

  2

1/2
β (X ) =
2  2Y ( X ) 
π  X 
91
and the function κ is defined as follows
κ (α ) =
1 + (α − 1) sin(πα )
2α (1 − α )
(3.5)
When all these expressions are substituted into the formula for the vertical
component of the displacement within the cohesive zone associated with
a fractal crack (3.3), and when the “final stretch” criterion for the subcritical crack (2.6) is applied within the restrictions of the Barenblatt’s condition R<<a, the following differential equation results
dR
1
1 1


=
 M ( ρ ) − − ln [ 4 ρ N (α , X , Y ) R / Rini ]
da N (α , X , Y ) 
2 2

(3.6)
Numerical integration of this equation yields the material resistance curves
R = R(a) and the beta-curves shown respectively in Figures 10 and 11.
The plots of stability indices corresponding to each value of the exponent α
are shown in Fig. 12. All figures have been drawn at the ductility index ρ =
20 and the initial crack length X 0 = 10. Finally, diagrams depicted in Fig.
13 show the crack length as a function of time in an analogous way to the
results presented in Fig. 4 (for cracks in viscoelastic media) and in Fig. 9
(for smooth cracks embedded in ductile solids).
EFFECTIVE MATERIAL RESISTANCE, Y=R/Rini
2.5
α =0.40
2
α =0.45
α =0.50
1.5
1
10
11
12
13
14
Fig. 10. Material resistance curves R(a)/R ini obtained for a smooth crack (the lowest curve, α = 0.5) and two fractal cracks defined by the fractal exponent α equal
0.45 (or D = 1.1) and α = 0.40 (or D = 1.2). It is noted that increasing roughness
of the crack surfaces, measured either by α, or the dimension D, enhances the effects of the slow crack growth on the effective material resistance. When the effective material resistance is compared with the one obtained for a smooth crack, one
observes 57.4% increase for the fractal crack described by α = 0.40 and 26.6% for
the fractal crack with α = 0.45.
92
APPLIED LOAD, β=σ/σY
0.36
α =0.40
0.34
α =0.45
0.32
0.3
α =0.50
14
13
12
11
10
Fig. 11. Applied load shown as function of the current crack length. The lowest
curve corresponds to a smooth crack, while the other two describe fractal cracks
with rough surfaces. Degree of fractality is determined by the exponent α or the
dimension D; for the intermediate curve α = 0.45 (or D = 1.1), while for the top
curve α = 0.40 (or D = 1.2). Enhancement of the critical load compared to that of
the smooth crack attains 20.8% for fractal with α = 0.40 and 8.9% for fractal with
α = 0.45.
0.01
−3
STABILITY INDEX, S
5×10
0
α =0.50
−3
α =0.45
α =0.40
− 5×10
− 0.01
11
12
13
14
Fig. 12. Stability indices shown as functions of the current crack length for
a smooth crack and two fractal (rough) cracks. Intersection points of the S-curves
with the horizontal line drawn at S = 0 indicate the location of the terminal instability states resulting for a given degree of fractality. Enhancement in the terminal
crack length compared to the result valid for a smooth crack is 7.77% for a fractal
crack described by α = 0.40 and 5.53% for a fractal with α = 0.45.
93
14
13
12
α =0.50
α =0.45
α =0.40
11
10
0.28
0.3
0.32
0.34
0.36
0.38
NONDIMENSIONAL TIME
Fig. 13. Crack length shown as a function of nondimensional time parameter for
a smooth crack (α = 0.5) and two fractal cracks (α = 0.45 and 0.40). It is seen
that the increased roughness of the crack surfaces leads to a more pronounced
quasi-static crack growth. Onset of growth process occurs at a certain threshold of
the applied load β min = 0.285, and it continues until the slopes of the curves approach infinity.
It is seen that the effect of the roughness of the crack surfaces on the
process of slow stable crack growth is substantial. Rougher surfaces of
a propagating crack tend to enhance the process of the slow stable crack
growth, which precedes onset of the catastrophic fracture.
4. Conclusions
Effects of two parameters on enhancement of the time-dependent fracture
manifested by a slow stable crack propagation that precedes catastrophic
failure in ductile materials have been studied. One of these parameters is
related to the material ductility (ρ) and the other describes the geometry
(roughness) of crack surface and is measured by the degree of fractality
represented by the fractal exponent α, or – equivalently – by the fractal
dimension D. These studies of early stages of ductile fracture were
ceded by a brief summary of modeling of the phenomenon of delayed fracture in polymeric materials, sometimes referred to as “creep rupture”. Despite different physical mechanisms involved in the preliminary stable
crack extension and despite different mathematical representations,
a remarkable similarity of the end results pertaining to the two
na of slow crack growth (SCG) that occur either in viscoelastic or in ductile
media has been demonstrated. For the viscoelastic material the response
to the deformation and fracture processes consists in the time-dependent
nature of the constitutive equations that play the dominant role in deter-
94
mination of the stable crack extension. For the ductile materials, even
though there is no explicit time-dependence in the first principles that govern behavior of these solids, the redistribution of plastic strains in the region adjacent to the front of a propagating crack enables quasi-static continuing crack growth. It has been shown that this process is very similar to
a “creeping crack” that propagates through a polymer.
Figures 14a and 14b illustrate the three ranges of crack growth,
namely
I, Region of no growth,
II, Region of stable crack extension, and
III, Region of unstable propagation.
LOADING PARAMETER, Q=πσ/2σY
dQ
0
da
Qf
dQ
dQ
 0
da
dQ
0
da

Q0
a0
da
af
Fig. 14a. Phases of crack development in a thick-wall welded pressure vessel
LOADING PARAMETER, Q=πσ/2σY
Qf
Qi
a0
af
Fig. 14b. Three ranges of crack growth in ductile solids: I – no growth region, II –
stable quasi-static growth range of load, III – unstable growth (catastrophic fracture)
95
Clearly, the existence of the incubation period followed by the propagation
phase for a crack embedded in a viscoelastic medium resembles those
three growth stages. Our study indicates that both material ductility and
geometrical irregularities, such as roughness of the crack surface, enhance
the period of slow stable crack extension and substantially influence the
characteristics of the terminal instability state attained at the end of the
slow crack growth process. For the purpose of fracture prevention both
ductility and crack surface roughness are desirable properties.
APPENDIX
Delayed fracture occurring in a linearly viscoelastic solid such as the one
discussed in Section 1, consists of two distinct stages: (1) incubation
phase, during which the opening displacement associated with the crack
increases in time, but the crack remains stationary, and (2) propagation
phase, when the crack advances up to the critical length (Griffith length),
at which transition to unstable crack extension takes place. Stage I (incubation) is described by the Wnuk-Knauss equation (1.5) and for the standard linear solid (see Fig. 2) the predicted duration of the incubation phase
t 1 is given as

β1 
t1 = τ 2 ln 

 1 + β1 − n 
(A.1)
The phase II (crack propagation) is governed by the Mueller-KnaussSchapery equation (1.6). For the nondimensional creep compliance function Ψ (t) defined by (1.11) the resulting equation of motion, which relates crack length x to time t, is given by (1.22), while the duration of the
propagation phase is predicted as follows
 β1n 
1 + β1 − n  
 τ   n
=
t2  2  
ln 
 + ln 

 δ  1 + β1 1 + β1 − n 
 β1  
(A.2)
The total life time T cr of the component manufactured of a polymeric material that obeys the constitutive equations described in Section 1 is obtained as the sum of (A.1) and (A.2), namely

 β1n 
1 + β1 − n  
β1   τ 2   n
Tcr = t1 + t2 =τ 2 ln 
ln 
+ 
 + ln 
  (A.3)
 1 + β1 − n   δ  1 + β1 1 + β1 − n 
 β1  
For Solithane 50/50, a polymer which is used to model mechanical properties of the solid rocket fuel, the times t 1 , t 2 and T cr were evaluated by
Knauss (1969) and Mohanty (1972). The moduli E 1 and E 2 and the viscosity η 2 involved in the standard linear solid that was applied in these studies
are as follows
96
E1 = 6.65*103 lb / in 2
(A.4)
E2 = 3.69*103 lb / in 2
η2 = 1.36*103 sec lb / in 2
This leads to β 1 = 1.8, the relaxation time τ 2 = η 2 /E 2 =0.368 sec and the
maximum crack length quotient n max = 1+ β 1 = 2.8. The structural length
Δ was estimated as 4.5x10-4 inch, while the pre-cut cracks used in the experiments were on the order of 0.225 inch. This yielded the inner structural constant δ = 2x10-3. From (A.4) the “glassy” and the “rubbery” values of
the creep compliance function can be readily calculated, namely
= J=
J glassy
(0) 1.50*10−4 in 2 / lb
J rubbery = J (∞) = 4.22*10−4 in 2 / lb
(A.5)
For detailed calculations the reader is referred to Knauss (1969) and Mohanty (1972).
The glassy (instantaneous) and rubbery (upon complete relaxation) compliance function values, as given in (A.5), allow one to establish the domains of the delayed fracture, such as “no growth”, incubation or the
propagation domains. It should be noted that the creep compliance functions involved in these experimental investigations were obtained by use
of the Staverman and Schwarzl (1953) method, see also Halaunbrenner
and Kubisz (1968).
In general, the propagation of a crack embedded in the viscoelastic medium will occur within a certain range of applied load. The two limiting values are (1) the Griffith stress evaluated for the initial crack size a 0 , which
is



σG = 



2 Eγ
π a0
(A.6)
K IC
π a0
and (2) the propagation threshold stress
=
σ threshold
J (0)
=
σG
J (∞ )
J glassy
J rubbery
σG
(A.7)
For the standard linear solid expression (A.7) reads
σ threshold =
1
σG
1 + β1
(A.8)
97
Using these relations one can predict the range of the applied loads for
a successful delayed fracture test performed on Solithane 50/50 as being
between 6/10 of the Griffith stress and the Griffith stress itself.
Summarizing, for the loads below the threshold stress given in (A.7) and
(A.8) one enters the “no growth” domain, where propagation does not
take place and the cracks in this region remain dormant. The other extreme is attained when the applied constant stress σ 0 reaches the Griffith
level σ G . When σ 0 approaches the Griffith stress we observe an instantaneous fracture as in a brittle medium with no delay effects. Therefore, one
may conclude that the delayed fracture occurs only in the range
σ threshold ≤ σ 0 ≤ σ G
σG
≤ σ0 ≤ σG
1 + β1
(A.9)
The second expression in (A.9) pertains to the standard linear model.
Let us now consider a numerical example for a polymer characterized by
the following properties β 1 = 10, τ 2 = 1 sec and δ = 10-4. Pertinent calculations are performed for three levels of the applied load, measured either
by the crack length quotient n (= σ G 2/σ 0 2) or by the load ratio s = σ 0 /σ G ,
namely n = 8.16 (s =0.35), n = 6.25 (s = 0.40) and n = 4 (s = 0.50). Applying (A.1) and (A.2) we obtain the following incubation (t 1 ) and time-tofailure (t 2 ) values
=
=
n 8.16;
s 0.35
t1 = 1.26sec
−4
=
=
t2 (1/10
)(0.277) sec 46.2 min
=
=
n 6.25;
s 0.40
t1 = 0.744sec
(A.10)
−4
=
=
t2 (1/10
)(0.720) sec 120 min
=
n 4;=
s .50
t1 = 0.375sec
−4
=
=
t2 (1/10
)(1.232) sec 205 min
It is noted that for this material the range of the applied stress for the delayed fracture to occur is contained within the interval (0.3 σ G , σ G ). For
applied stress less than the threshold stress of 0.3 σ G the phenomenon of
delayed fracture vanishes, and the crack remains stationary.
For ductile solids there are no time-dependent moduli present in the
stitutive equations. Yet, the process of quasi-static continuing crack
growth does manifest itself as “slow crack growth” (SCG), which in almost
all cases precedes the terminal instability state tantamount to the
strophic fracture. To understand this phenomenon it is essential to view
each instant in the crack growth process as a state of the equilibrium
maintained between the applied external effort, say the driving force G or
98
the Rice’s J-integral or the stress intensity factor K I , and the material resistance to crack propagation designated by the index “R”. In mathematical terms this statement reads
G (σ , a ) = GR (a )
(A.11)
J (σ , a ) = J R (a )
K I (σ , a ) = K R (a )
Both measures of the external effort G and J are defined in the well-known
manner; J = G = K I 2/E 1 , while the entities on the right hand sides of
(A.11) are defined by the governing equations (2.11) for a smooth crack
and by (3.6) for a fractal crack. According to Wells (1963) the J-criterion
for fracture may be replaced by an equivalent COD (or δ tip ) criterion – just
as it is predicted for the structured cohesive crack model, see equations
(2.16). In this way all expressions in (A.11) may be replaced by just a single relation
RAPPL (σ , a ) = RMAT (a )
(A.12)
For simplicity the symbol R MAT (a) is represented in Sections 2 and 3 by
R(a) – or in its nondimensional version – by Y(X). In this way the equilibrium length of the cohesive zone R serves as a measure of the external
effort
2
1  πσ 
R(σ , a ) = 
 a
2  2σ Y 
(A.13)
This is a well-known expression resulting for the small-scale yielding case
(when the Barenblatt’s condition, R<<a, holds) from the Dugdale model.
During the slow crack growth phase the quantity defined by (A.13) must
be equal R MAT defined by the governing differential equations, either (2.11)
for a smooth crack or (3.6) for a fractal crack. Attainment of the terminal
instability state requires that two conditions are satisfied simultaneously
RAPPL (σ , a ) = RMAT (a )
∂RAPPL (σ , a ) dRMAT (a )
=
da
∂a
(A.14)
It should be noted that the derivative in the second expression of (A.14) is
proportional to the second derivative of the total potential of the system,
namely
∂RAPPL (σ , a )
∂ 2 Π (σ , a )
−
∂a
∂a 2
(A.15)
Using (2.16) and recalling that the J-integral equals − d Π d (2a ) one can
readily provide a constant of proportionality between R APPL and the
J-integral and their derivatives, which appear in (A.15). The potential of
the cracked body Π(σ,a) is defined as follows
99
, a)
Π (σ=
1
σ ij ε ij dV − ∫ Ti ui dS − SE (a)
2 V∫
ST
(A.16)
Symbol SE(a) denotes the surface energy term introduced by Griffith. Using (A.13) we evaluate the derivative needed in (A.14)
∂RAPPL (σ , a ) ∂  1  πσ
 
=
∂a
∂a  2  2σ Y

2
2
  1  πσ 
RAPPL Y

a
=
=
=



a
X
  2  2σ Y 
(A.17)
At the terminal instability point this expression should equal the derivative
dY/dX defined in (2.11) and/or (3.6), namely
 dY 
Y 
= 
 dX 
 X transition
transition
(A.18)
The index “transition” refers to the attainment of the terminal instability
state, which is tantamount to the transition from stable to unstable crack
propagation. It is noted that the condition (A.18) is exactly equivalent to
the requirement that the stability index defined in (2.24) equals zero. Fig.
A1 illustrates how the condition (A.18) may be used to determine the state
of the terminal instability. The intersection points shown in Fig. A1 coincide
exactly with the results obtained in Section 2 for a smooth crack for three
different levels of the material ductility, compare Fig.8.
NONDIMENSIONAL SLOPES, ∂RAPPL/∂a and dRMAT/da
0.13
ρ =80
0.12
ρ =40
ρ =20
0.11
0.1
10
11
12
13
14
Fig. A1. The nearly straight lines depict the functional relationship between the
slope of the material R-curve, (dY/dX) MAT and the crack length X, while the other
set of curves represents the measure of externally applied effort (Y/X) APPL . Points
of intersection between these curves designate the terminal instability states,
compare Figure 8.
100
In an analogous way the case of the fractal crack can be resolved. Here
one has
f
f
∂RAPPL
RAPPL
RAPPL
Y
= =
N (α , X , Y ) =
N (α , X , Y )
∂a
∂a
a
X
f
dRMAT
dY
= N (α , X , Y )
da
dX
(A.19)
When these two entities representing the rate of the external effort and
the rate of material resistance to continuing crack extension, the factor
N(α,X,Y) cancels out, and one recovers the condition for the terminal instability expressed by (A.18).
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102
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103
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