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Casanova had dug up a freshly buried corpse to play a practical joke on an enemy and exact revenge, but the victim went into a paralysis, never to recover. And in another scandal, a young girl who had duped him accused him of rape and went to the officials. Casanova was later acquitted of this crime for lack of evidence, but by this time, he had already fled from Venice. Escaping to Parma, Casanova entered into a three-month affair with a Frenchwoman he named "Henriette", perhaps the deepest love he ever experienced—a woman who combined beauty, intelligence, and culture. In his words, "They who believe that a woman is incapable of making a man equally happy all the twenty-four hours of the day have never known an Henriette.
The joy which flooded my soul was far greater when I conversed with her during the day than when I held her in my arms at night. Having read a great deal and having natural taste, Henriette judged rightly of everything." She also judged Casanova astutely. As noted Casanovist J. Rives Childs wrote: Grand tour Crestfallen and despondent, Casanova returned to Venice, and after a good gambling streak, he recovered and set off on a grand tour, reaching Paris in 1750. Along the way, from one town to another, he got into sexual escapades resembling operatic plots. In Lyon, he entered the society of Freemasonry, which appealed to his interest in secret rites and which, for the most part, attracted men of intellect and influence who proved useful in his life, providing valuable contacts and uncensored knowledge.
Casanova was also attracted to Rosicrucianism. In Lyons, Casanova become companion and finally took the highest degree of Scottish Rite Master Mason. Regarding his initiation to the Scottish Rite Freemasonry in Lyons, the Memoirs said: Casanova stayed in Paris for two years, learned the language, spent much time at the theater, and introduced himself to notables. Soon, however, his numerous liaisons were noted by the Paris police, as they were in nearly every city he visited. In 1752, his brother Francesco and he moved from Paris to Dresden, where his mother and sister Maria Maddalena were living. His new play, La Moluccheide, now lost, was performed at the Royal Theatre, where his mother often played in lead roles.
He then visited Prague and Vienna, where the tighter moral atmosphere of the latter city was not to his liking. He finally returned to Venice in 1753. In Venice, Casanova resumed his escapades, picking up many enemies and gaining the greater attention of the Venetian inquisitors. His police record became a lengthening list of reported blasphemies, seductions, fights, and public controversy. A state spy, Giovanni Manucci, was employed to draw out Casanova's knowledge of cabalism and Freemasonry and to examine his library for forbidden books. Senator Bragadin, in total seriousness this time (being a former inquisitor himself), advised his "son" to leave immediately or face the stiffest consequences.
Imprisonment and escape On 26 July 1755, at age 30, Casanova was arrested for affront to religion and common decency: "The Tribunal, having taken cognizance of the grave faults committed by G. Casanova primarily in public outrages against the holy religion, their Excellencies have caused him to be arrested and imprisoned under the Leads." "The Leads" was a prison of seven cells on the top floor of the east wing of the Doge's palace, reserved for prisoners of higher status as well as certain types of offenders—such as political prisoners, defrocked or libertine priests or monks, and usurers—and named for the lead plates covering the palace roof.
The following 12 September, without a trial and without being informed of the reasons for his arrest and of the sentence, he was sentenced to five years imprisonment. He was placed in solitary confinement with clothing, a pallet bed, table, and armchair in "the worst of all the cells", where he suffered greatly from the darkness, summer heat, and "millions of fleas". He was soon housed with a series of cellmates, and after five months and a personal appeal from Count Bragadin, was given warm winter bedding and a monthly stipend for books and better food. During exercise walks he was granted in the prison garret, he found a piece of black marble and an iron bar which he smuggled back to his cell; he hid the bar inside his armchair.
When he was temporarily without cellmates, he spent two weeks sharpening the bar into a spike on the stone. Then he began to gouge through the wooden floor underneath his bed, knowing that his cell was directly above the Inquisitor's chamber. Just three days before his intended escape, during a festival when no officials would be in the chamber below, Casanova was moved to a larger, lighter cell with a view, despite his protests that he was perfectly happy where he was. In his new cell, "I sat in my armchair like a man in a stupor; motionless as a statue, I saw that I had wasted all the efforts I had made, and I could not repent of them.
I felt that I had nothing to hope for, and the only relief left to me was not to think of the future." Overcoming his inertia, Casanova set upon another escape plan. He solicited the help of the prisoner in the adjacent cell, Father Balbi, a renegade priest. The spike, carried to the new cell inside the armchair, was passed to the priest in a folio Bible carried under a heaping plate of pasta by the hoodwinked jailer. The priest made a hole in his ceiling, climbed across and made a hole in the ceiling of Casanova's cell. To neutralize his new cellmate, who was a spy, Casanova played on his superstitions and terrorized him into silence.
When Balbi broke through to Casanova's cell, Casanova lifted himself through the ceiling, leaving behind a note that quoted the 117th Psalm (Vulgate): "I shall not die, but live, and declare the works of the Lord". The spy remained behind, too frightened of the consequences if he were caught escaping with the others. Casanova and Balbi pried their way through the lead plates and onto the sloping roof of the Doge's Palace, with a heavy fog swirling. The drop to the nearby canal being too great, Casanova prised open the grate over a dormer window, and broke the window to gain entry.
They found a long ladder on the roof, and with the additional use of a bedsheet "rope" that Casanova had prepared, lowered themselves into the room whose floor was 25 feet below. They rested until morning, changed clothes, then broke a small lock on an exit door and passed into a palace corridor, through galleries and chambers, and down stairs, where by convincing the guard they had inadvertently been locked into the palace after an official function, they left through a final door. It was 6:00 in the morning and they escaped by gondola. Eventually, Casanova reached Paris, where he arrived on the same day (5 January 1757) that Robert-François Damiens made an attempt on the life of Louis XV.
(Casanova would later witness and describe his execution.) Thirty years later in 1787, Casanova wrote Story of My Flight, which was very popular and was reprinted in many languages, and he repeated the tale a little later in his memoirs. Casanova's judgment of the exploit is characteristic: Return to Paris He knew his stay in Paris might be a long one and he proceeded accordingly: "I saw that to accomplish anything I must bring all my physical and moral faculties in play, make the acquaintance of the great and the powerful, exercise strict self-control, and play the chameleon." Casanova had matured, and this time in Paris, though still depending at times on quick thinking and decisive action, he was more calculating and deliberate.
His first task was to find a new patron. He reconnected with old friend de Bernis, now the Foreign Minister of France. Casanova was advised by his patron to find a means of raising funds for the state as a way to gain instant favor. Casanova promptly became one of the trustees of the first state lottery, and one of its best ticket salesmen. The enterprise earned him a large fortune quickly. With money in hand, he traveled in high circles and undertook new seductions. He duped many socialites with his occultism, particularly the Marquise Jeanne d'Urfé, using his excellent memory which made him appear to have a sorcerer's power of numerology.
In Casanova's view, "deceiving a fool is an exploit worthy of an intelligent man". Casanova claimed to be a Rosicrucian and an alchemist, aptitudes which made him popular with some of the most prominent figures of the era, among them Madame de Pompadour, Count de Saint-Germain, d'Alembert, and Jean-Jacques Rousseau. So popular was alchemy among the nobles, particularly the search for the "philosopher's stone", that Casanova was highly sought after for his supposed knowledge, and he profited handsomely. He met his match, however, in the Count de Saint-Germain: "This very singular man, born to be the most barefaced of all imposters, declared with impunity, with a casual air, that he was three hundred years old, that he possessed the universal medicine, that he made anything he liked from nature, that he created diamonds."
De Bernis decided to send Casanova to Dunkirk on his first spying mission. Casanova was paid well for his quick work and this experience prompted one of his few remarks against the ancien régime and the class on which he was dependent. He remarked in hindsight, "All the French ministers are the same. They lavished money which came out of the other people's pockets to enrich their creatures, and they were absolute: The down-trodden people counted for nothing, and, through this, the indebtedness of the State and the confusion of finances were the inevitable results. A Revolution was necessary." As the Seven Years' War began, Casanova was again called to help increase the state treasury.
He was entrusted with a mission of selling state bonds in Amsterdam, Holland being the financial center of Europe at the time. He succeeded in selling the bonds at only an 8% discount, and the following year was rich enough to found a silk manufactory with his earnings. The French government even offered him a title and a pension if he would become a French citizen and work on behalf of the finance ministry, but he declined, perhaps because it would frustrate his Wanderlust. Casanova had reached his peak of fortune, but could not sustain it. He ran the business poorly, borrowed heavily trying to save it, and spent much of his wealth on constant liaisons with his female workers who were his "harem".
For his debts, Casanova was imprisoned again, this time at For-l'Évêque, but was liberated four days afterwards, upon the insistence of the Marquise d'Urfé. Unfortunately, though he was released, his patron de Bernis was dismissed by Louis XV at that time and Casanova's enemies closed in on him. He sold the rest of his belongings and secured another mission to Holland to distance himself from his troubles. On the run This time, however, his mission failed and he fled to Cologne, then Stuttgart in the spring of 1760, where he lost the rest of his fortune. He was yet again arrested for his debts, but managed to escape to Switzerland.
Weary of his wanton life, Casanova visited the monastery of Einsiedeln and considered the simple, scholarly life of a monk. He returned to his hotel to think on the decision, only to encounter a new object of desire, and reverting to his old instincts, all thoughts of a monk's life were quickly forgotten. Moving on, he visited Albrecht von Haller and Voltaire, and arrived in Marseille, then Genoa, Florence, Rome, Naples, Modena, and Turin, moving from one sexual romp to another. In 1760, Casanova started styling himself the Chevalier de Seingalt, a name he would increasingly use for the rest of his life.
On occasion, he would also call himself Count de Farussi (using his mother's maiden name) and when Pope Clement XIII presented Casanova with the Papal Order of the Éperon d'or, he had an impressive cross and ribbon to display on his chest. Back in Paris, he set about one of his most outrageous schemes—convincing his old dupe the Marquise d'Urfé that he could turn her into a young man through occult means. The plan did not yield Casanova the big payoff he had hoped for, and the Marquise d'Urfé finally lost faith in him. Casanova traveled to England in 1763, hoping to sell his idea of a state lottery to English officials.
He wrote of the English, "the people have a special character, common to the whole nation, which makes them think they are superior to everyone else. It is a belief shared by all nations, each thinking itself the best. And they are all right." Through his connections, he worked his way up to an audience with King George III, using most of the valuables he had stolen from the Marquise d'Urfé. While working the political angles, he also spent much time in the bedroom, as was his habit. As a means to find females for his pleasure, not being able to speak English, he put an advertisement in the newspaper to let an apartment to the "right" person.
He interviewed many young women, choosing one "Mistress Pauline" who suited him well. Soon, he established himself in her apartment and seduced her. These and other liaisons, however, left him weak with venereal disease and he left England broke and ill. He went on to the Austrian Netherlands, recovered, and then for the next three years, traveled all over Europe, covering about 4,500 miles by coach over rough roads, and going as far as Moscow and Saint Petersburg (the average daily coach trip being about 30 miles). Again, his principal goal was to sell his lottery scheme to other governments and repeat the great success he had with the French government, but a meeting with Frederick the Great bore no fruit and in the surrounding German lands, the same result.
Not lacking either connections or confidence, Casanova went to Russia and met with Catherine the Great, but she flatly turned down the lottery idea. In 1766, he was expelled from Warsaw following a pistol duel with Colonel Franciszek Ksawery Branicki over an Italian actress, a lady friend of theirs. Both duelists were wounded, Casanova on the left hand. The hand recovered on its own, after Casanova refused the recommendation of doctors that it be amputated. From Warsaw, he traveled to Breslau in the Kingdom of Prussia, then to Dresden, where he contracted yet another venereal infection. He returned to Paris for several months in 1767 and hit the gambling salons, only to be expelled from France by order of Louis XV himself, primarily for Casanova's scam involving the Marquise d'Urfé.
Now known across Europe for his reckless behavior, Casanova would have difficulty overcoming his notoriety and gaining any fortune, so he headed for Spain, where he was not as well known. He tried his usual approach, leaning on well-placed contacts (often Freemasons), wining and dining with nobles of influence, and finally arranging an audience with the local monarch, in this case Charles III. When no doors opened for him, however, he could only roam across Spain, with little to show for it. In Barcelona, he escaped assassination and landed in jail for 6 weeks. His Spanish adventure a failure, he returned to France briefly, then to Italy.
Return to Venice In Rome, Casanova had to prepare a way for his return to Venice. While waiting for supporters to gain him legal entry into Venice, Casanova began his modern Tuscan-Italian translation of the Iliad, his History of the Troubles in Poland, and a comic play. To ingratiate himself with the Venetian authorities, Casanova did some commercial spying for them. After months without a recall, however, he wrote a letter of appeal directly to the Inquisitors. At last, he received his long-sought permission and burst into tears upon reading "We, Inquisitors of State, for reasons known to us, give Giacomo Casanova a free safe-conduct ... empowering him to come, go, stop, and return, hold communication wheresoever he pleases without let or hindrance.
So is our will." Casanova was permitted to return to Venice in September 1774 after 18 years of exile. At first, his return to Venice was a cordial one and he was a celebrity. Even the Inquisitors wanted to hear how he had escaped from their prison. Of his three bachelor patrons, however, only Dandolo was still alive and Casanova was invited back to live with him. He received a small stipend from Dandolo and hoped to live from his writings, but that was not enough. He reluctantly became a spy again for Venice, paid by piece work, reporting on religion, morals, and commerce, most of it based on gossip and rumor he picked up from social contacts.
He was disappointed. No financial opportunities of interest came about and few doors opened for him in society as in the past. At age 49, the years of reckless living and the thousands of miles of travel had taken their toll. Casanova's smallpox scars, sunken cheeks, and hook nose became all the more noticeable. His easygoing manner was now more guarded. Prince Charles de Ligne, a friend (and uncle of his future employer), described him around 1784: Venice had changed for him. Casanova now had little money for gambling, few willing females worth pursuing, and few acquaintances to enliven his dull days.
He heard of the death of his mother and, more paining, visited the deathbed of Bettina Gozzi, who had first introduced him to sex and who died in his arms. His Iliad was published in three volumes, but to limited subscribers and yielding little money. He got into a published dispute with Voltaire over religion. When he asked, "Suppose that you succeed in destroying superstition. With what will you replace it?" Voltaire shot back, "I like that. When I deliver humanity from a ferocious beast which devours it, can I be asked what I shall put in its place." From Casanova's point of view, if Voltaire had "been a proper philosopher, he would have kept silent on that subject ... the people need to live in ignorance for the general peace of the nation".
In 1779, Casanova found Francesca, an uneducated seamstress, who became his live-in lover and housekeeper, and who loved him devotedly. Later that year, the Inquisitors put him on the payroll and sent him to investigate commerce between the papal states and Venice. Other publishing and theater ventures failed, primarily from lack of capital. In a downward spiral, Casanova was expelled again from Venice in 1783, after writing a vicious satire poking fun at Venetian nobility. In it, he made his only public statement that Grimani was his true father. Forced to resume his travels again, Casanova arrived in Paris, and in November 1783 met Benjamin Franklin while attending a presentation on aeronautics and the future of balloon transport.
For a while, Casanova served as secretary and pamphleteer to Sebastian Foscarini, Venetian ambassador in Vienna. He also became acquainted with Lorenzo Da Ponte, Mozart's librettist, who noted about Casanova, "This singular man never liked to be in the wrong." Notes by Casanova indicate that he may have made suggestions to Da Ponte concerning the libretto for Mozart's Don Giovanni. Final years in Bohemia In 1785, after Foscarini died, Casanova began searching for another position. A few months later, he became the librarian to Count Joseph Karl von Waldstein, a chamberlain of the emperor, in the Castle of Dux, Bohemia (now in the Czech Republic).
The Count—himself a Freemason, cabalist, and frequent traveler—had taken to Casanova when they had met a year earlier at Foscarini's residence. Although the job offered security and good pay, Casanova describes his last years as boring and frustrating, though it was the most productive time for writing. His health had deteriorated dramatically, and he found life among peasants to be less than stimulating. He was only able to make occasional visits to Vienna and Dresden for relief. Although Casanova got on well with the Count, his employer was a much younger man with his own eccentricities. The Count often ignored him at meals and failed to introduce him to important visiting guests.
Moreover, Casanova, the testy outsider, was thoroughly disliked by most of the other inhabitants of the Castle of Dux. Casanova's only friends seemed to be his fox terriers. In despair, Casanova considered suicide, but instead decided that he must live on to record his memoirs, which he did until his death. He visited Prague, the capital city and principal cultural center of Bohemia, on many occasions. In October 1787, he met Lorenzo da Ponte, the librettist of Wolfgang Amadeus Mozart's opera Don Giovanni, in Prague at the time of the opera's first production and likely met the composer, as well, at the same time.
There is reason to believe that he was also in Prague in 1791 for the coronation of Holy Roman Emperor Leopold II as king of Bohemia, an event that included the first production of Mozart's opera La clemenza di Tito. Casanova is known to have drafted dialogue suitable for a Don Juan drama at the time of his visit to Prague in 1787, but none of his verses were ever incorporated into Mozart's opera. His reaction to seeing licentious behavior similar to his own held up to moral scrutiny as it is in Mozart's opera is not recorded. In 1797, word arrived that the Republic of Venice had ceased to exist and that Napoleon Bonaparte had seized Casanova's home city.
It was too late to return home. Casanova died on 4 June 1798 at the age of 73. His last words are said to have been "I have lived as a philosopher and I die as a Christian". Casanova was buried at Dux (nowadays Duchcov in the Czech Republic), but the exact place of his grave was forgotten over the years, and remains unknown today. Memoirs The isolation and boredom of Casanova's last years enabled him to focus without distractions on his Histoire de ma vie, without which his fame would have been considerably diminished, if not blotted out entirely.
He began to think about writing his memoirs around 1780 and began in earnest by 1789, as "the only remedy to keep from going mad or dying of grief". The first draft was completed by July 1792, and he spent the next six years revising it. He puts a happy face on his days of loneliness, writing in his work, "I can find no pleasanter pastime than to converse with myself about my own affairs and to provide a most worthy subject for laughter to my well-bred audience." His memoirs were still being compiled at the time of his death, his account having reached only the summer of 1774.
A letter by him in 1792 states that he was reconsidering his decision to publish them, believing that his story was despicable and he would make enemies by writing the truth about his affairs. But he decided to proceed, using initials instead of actual names and toning down the strongest passages. He wrote in French instead of Italian because "the French language is more widely known than mine". The memoirs open with: Casanova wrote about the purpose of his book: He also advises his readers that they "will not find all my adventures. I have left out those which would have offended the people who played a part in them, for they would cut a sorry figure in them.
Even so, there are those who will sometimes think me too indiscreet; I am sorry for it." And in the final chapter, the text abruptly breaks off with hints at adventures unrecorded: "Three years later I saw her in Padua, where I resumed my acquaintance with her daughter on far more tender terms." In their original publication, the memoirs were divided into twelve volumes, and the unabridged English translation by Willard R. Trask runs to more than 3,500 pages. Though his chronology is at times confusing and inaccurate, and many of his tales exaggerated, much of his narrative and many details are corroborated by contemporary writings.
He has a good ear for dialogue and writes at length about all classes of society. Casanova, for the most part, is candid about his faults, intentions, and motivations, and shares his successes and failures with good humor. The confession is largely devoid of repentance or remorse. He celebrates the senses with his readers, especially regarding music, food, and women. "I have always liked highly seasoned food. ... As for women, I have always found that the one I was in love with smelled good, and the more copious her sweat the sweeter I found it." He mentions over 120 adventures with women and girls, with several veiled references to male lovers as well.
He describes his duels and conflicts with scoundrels and officials, his entrapments and his escapes, his schemes and plots, his anguish and his sighs of pleasure. He demonstrates convincingly, "I can say vixi ('I have lived')." The manuscript of Casanova's memoirs was held by his relatives until it was sold to F. A. Brockhaus publishers, and first published in heavily abridged versions in German around 1822, then in French. During World War II, the manuscript survived the allied bombing of Leipzig. The memoirs were heavily pirated through the ages and have been translated into some twenty languages. But not until 1960 was the entire text published in its original language of French.
In 2010 the manuscript was acquired by the National Library of France, which has started digitizing it. Relationships For Casanova, as well as his contemporary sybarites of the upper class, love and sex tended to be casual and not endowed with the seriousness characteristic of the Romanticism of the 19th century. Flirtations, bedroom games, and short-term liaisons were common among nobles who married for social connections rather than love. Although multi-faceted and complex, Casanova's personality, as he described it, was dominated by his sensual urges: "Cultivating whatever gave pleasure to my senses was always the chief business of my life; I never found any occupation more important.
Feeling that I was born for the sex opposite of mine, I have always loved it and done all that I could to make myself loved by it." He noted that he sometimes used "assurance caps" to prevent impregnating his mistresses. Casanova's ideal liaison had elements beyond sex, including complicated plots, heroes and villains, and gallant outcomes. In a pattern he often repeated, he would discover an attractive woman in trouble with a brutish or jealous lover (Act I); he would ameliorate her difficulty (Act II); she would show her gratitude; he would seduce her; a short exciting affair would ensue (Act III); feeling a loss of ardor or boredom setting in, he would plead his unworthiness and arrange for her marriage or pairing with a worthy man, then exit the scene (Act IV).
As William Bolitho points out in Twelve Against the Gods, the secret of Casanova's success with women "had nothing more esoteric in it than [offering] what every woman who respects herself must demand: all that he had, all that he was, with (to set off the lack of legality) the dazzling attraction of the lump sum over what is more regularly doled out in a lifetime of installments." Casanova advises, "There is no honest woman with an uncorrupted heart whom a man is not sure of conquering by dint of gratitude. It is one of the surest and shortest means."
Alcohol and violence, for him, were not proper tools of seduction. Instead, attentiveness and small favors should be employed to soften a woman's heart, but "a man who makes known his love by words is a fool". Verbal communication is essential—"without speech, the pleasure of love is diminished by at least two-thirds"—but words of love must be implied, not boldly proclaimed. Despite detailing what was clearly an abduction and gang rape ("It was during one Carnival, midnight had struck, we were eight, all masked, roving through the city ..."), Casanova convinced himself that the victim was willing. By his own account, mutual consent was important, but he avoided easy conquests or overly difficult situations as not suitable for his purposes.
He strove to be the ideal escort in the first act—witty, charming, confidential, helpful—before moving into the bedroom in the third act. Casanova claims not to be predatory ("my guiding principle has been never to direct my attack against novices or those whose prejudices were likely to prove an obstacle"); however, his conquests did tend to be insecure or emotionally exposed women. Casanova valued intelligence in a woman: "After all, a beautiful woman without a mind of her own leaves her lover with no resource after he had physically enjoyed her charms." His attitude towards educated women, however, was an unfavorable one: "In a woman learning is out of place; it compromises the essential qualities of her sex ... no scientific discoveries have been made by women ... (which) requires a vigor which the female sex cannot have.
But in simple reasoning and in delicacy of feeling we must yield to women." Casanova writes that he stopped short of intercourse with a 13-year-old named Helene: "little Helene, whom I enjoyed, while leaving her intact." In 1765, when he was 40, he purchased a 12-year-old girl in St. Petersburg as a sexual slave. In the memoirs, he described the Russian girl as emphatically prepubescent: "Her breasts had still not finished budding. She was in her thirteenth year. She had nowhere the definitive mark of puberty." (III, 196–7; X, 116–17). In 1774, when he was almost 50, Casanova encountered in Trieste a former lover, the actress Irene, now accompanied by her nine-year-old daughter.
"A few days later she came, with her daughter, who pleased me (qui me plut) and who did not reject my caresses. One fine day, she met with Baron Pittoni, who loved little girls as much as I did (aimant autant que moi les petites filles), and took a liking to Irene’s girl, and asked the mother to do him the same honor some time that she had done to me. I encouraged her to receive the offer, and the baron fell in love. This was lucky for Irene." (XII, 238). Gambling Gambling was a common recreation in the social and political circles in which Casanova moved.
In his memoirs, Casanova discusses many forms of 18th-century gambling—including lotteries, faro, basset, piquet, biribi, primero, quinze, and whist—and the passion for it among the nobility and the high clergy. Cheats (known as "correctors of fortune") were somewhat more tolerated than today in public casinos and in private games for invited players, and seldom caused affront. Most gamblers were on guard against cheaters and their tricks. Scams of all sorts were common, and Casanova was amused by them. Casanova gambled throughout his adult life, winning and losing large sums. He was tutored by professionals, and he was "instructed in those wise maxims without which games of chance ruin those who participate in them".
He was not above occasionally cheating and at times even teamed with professional gamblers for his own profit. Casanova claims that he was "relaxed and smiling when I lost, and I won without covetousness". However, when outrageously duped himself, he could act violently, sometimes calling for a duel. Casanova admits that he was not disciplined enough to be a professional gambler: "I had neither prudence enough to leave off when fortune was adverse, nor sufficient control over myself when I had won." Nor did he like being considered as a professional gambler: "Nothing could ever be adduced by professional gamblers that I was of their infernal clique."
Although Casanova at times used gambling tactically and shrewdly—for making quick money, for flirting, making connections, acting gallantly, or proving himself a gentleman among his social superiors—his practice also could be compulsive and reckless, especially during the euphoria of a new sexual affair. "Why did I gamble when I felt the losses so keenly? What made me gamble was avarice. I loved to spend, and my heart bled when I could not do it with money won at cards." Fame and influence Casanova was recognized by his contemporaries as an extraordinary person, a man of far-ranging intellect and curiosity. Casanova has been recognized by posterity as one of the foremost chroniclers of his age.
He was a true adventurer, traveling across Europe from end to end in search of fortune, seeking out the most prominent people of his time to help his cause. He was a servant of the establishment and equally decadent as his times, but also a participant in secret societies and a seeker of answers beyond the conventional. He was religious, a devout Catholic, and believed in prayer: "Despair kills; prayer dissipates it; and after praying man trusts and acts." Along with prayer he also believed in free will and reason, but clearly did not subscribe to the notion that pleasure-seeking would keep him from heaven.
He was, by vocation and avocation, a lawyer, clergyman, military officer, violinist, con man, pimp, gourmand, dancer, businessman, diplomat, spy, politician, medic, mathematician, social philosopher, cabalist, playwright, and writer. He wrote over twenty works, including plays and essays, and many letters. His novel Icosameron is an early work of science fiction. Born of actors, he had a passion for the theater and for an improvised, theatrical life. But with all his talents, he frequently succumbed to the quest for pleasure and sex, often avoiding sustained work and established plans, and got himself into trouble when prudent action would have served him better.
His true occupation was living largely on his quick wits, steely nerves, luck, social charm, and the money given to him in gratitude and by trickery. Prince Charles de Ligne, who understood Casanova well, and who knew most of the prominent individuals of the age, thought Casanova the most interesting man he had ever met: "there is nothing in the world of which he is not capable." Rounding out the portrait, the Prince also stated: "Casanova", like "Don Juan", is a long established term in the English language. According to Merriam Webster's Collegiate Dictionary, 11th ed., the noun Casanova means "Lover; esp: a man who is a promiscuous and unscrupulous lover".
The first usage of the term in written English was around 1852. References in culture to Casanova are numerous—in books, films, theater, and music. Works 1752 – Zoroastro: Tragedia tradotta dal Francese, da rappresentarsi nel Regio Elettoral Teatro di Dresda, dalla compagnia de' comici italiani in attuale servizio di Sua Maestà nel carnevale dell'anno MDCCLII. Dresden. 1753 – La Moluccheide, o Sia i gemelli rivali. Dresden. 1769 – Confutazione della Storia del Governo Veneto d'Amelot de la Houssaie. Lugano. 1772 – Lana caprina: Epistola di un licantropo. Bologna. 1774 – Istoria delle turbolenze della Polonia. Gorizia. 1775–78 – Dell'Iliade di Omero tradotta in ottava rima.
Venice. 1779 – Scrutinio del libro Eloges de M. de Voltaire par différents auteurs. Venice. 1780 – Opuscoli miscellanei (containing Duello a Varsavia and Lettere della nobil donna Silvia Belegno alla nobil donzella Laura Gussoni). Venice. 1780–81 – Le messager de Thalie. Venice. 1782 – Di aneddoti viniziani militari ed amorosi del secolo decimoquarto sotto i dogadi di Giovanni Gradenigo e di Giovanni Dolfin. Venice. 1783 – Né amori né donne, ovvero La stalla ripulita. Venice. 1786 – Soliloque d'un penseur. Prague. 1787 – Icosaméron, ou Histoire d'Édouard et d'Élisabeth qui passèrent quatre-vingts un ans chez les Mégamicres, habitants aborigènes du Protocosme dans l'intérieur de nôtre globe.
Prague. 1788 – Histoire de ma fuite des prisons de la République de Venise qu'on appelle les Plombs. Leipzig. 1790 – Solution du probléme deliaque. Dresden. 1790 – Corollaire à la duplication de l'hexaèdre. Dresden. 1790 – Démonstration géometrique de la duplication du cube. Dresden. 1797 – A Léonard Snetlage, docteur en droit de l'Université de Goettingue, Jacques Casanova, docteur en droit de l'Universitè de Padou. Dresden. 1822–29 – First edition of the Histoire de ma vie, in an adapted German translation in 12 volumes, as Aus den Memoiren des Venetianers Jacob Casanova de Seingalt, oder sein Leben, wie er es zu Dux in Böhmen niederschrieb.
The first full edition of the original French manuscript was not published until 1960, by Brockhaus (Wiesbaden) and Plon (Paris).
In popular culture Film Casanova (1918), a Hungarian film featuring Béla Lugosi The Loves of Casanova, or Casanova, a 1927 French film starring Ivan Mozzhukhin Il cavaliere misterioso (The Mysterious Rider), a 1948 film by Riccardo Freda, in which Casanova is played by Vittorio Gassman in his debut as a lead actor Poslední růže od Casanovy (The Last Rose from Casanova), a 1966 Czech film featuring Felix le Breux as aging Casanova during his stay at Duchcov Giacomo Casanova: Childhood and Adolescence, a 1969 feature film by Luigi Comencini, starring Leonard Whiting Fellini's Casanova, a 1976 feature film by Federico Fellini, starring Donald Sutherland La Nuit de Varennes (1982), a film featuring Marcello Mastroianni Casanova (1987), a television movie, starring Richard Chamberlain Le Retour de Casanova (1992), a French comedy starring Alain Delon Casanova (2005), a feature film featuring Heath Ledger, Sienna Miller and Charlie Cox Casanova Variations (2014), a feature film starring John Malkovich Zoroastro, Io Casanova (2017) an Italian film featuring Galatea Ranzi Music Casanova Fantasy Variations for Three Celli (1985), a piece for cello trio by Walter Burle Marx Casanova, (1986), a song by the Russian rock group Nautilus Pompilius.
Music by Vyacheslav Butusov, text by Ilya Kormil'tsev. "Casanova" (1987) song by R&B group LeVert. The song reached number 1 on the R&B chart as well as reaching number 5 on the pop chart.
Casanova (1996), an album by the UK chamber pop band The Divine Comedy, inspired by Casanova "Casanova 70" (1997), a single by French electronic duo Air Casanova (2000), a piece for cello and winds by Johan de Meij "Casanova in Hell" (2006), a song by the UK group Pet Shop Boys, from their album Fundamental Performance works Casanova (1923), a comic opera in three acts with prologue and epilogue, by Ludomir Różycki Casanova (1928), an operetta by Ralph Benatzky, based on music by Johann Strauss Jr. Camino Real (1953), a play by Tennessee Williams, in which an aging Casanova appears in a dream sequence Casanova's Homecoming (1985), an opera by Dominick Argento Casanova (2007), a play by Carol Ann Duffy and Told by an Idiot theatre company, starring Hayley Carmichael as a female Casanova Casanova (2008), a musical by Philip Godfrey, first performed at the Greenwich Playhouse, London Casanova (2016), a pasticcio opera by Julian Perkins and Stephen Pettitt, first performed in the Baroque Unwrapped series at Kings Place, London Casanova (2017), a ballet by Northern Ballet choreographed by Kenneth Tindall based on the biography by Ian Kelly.
Casanova (2019), a musical performed by Takarazuka Revue and starring Rio Asumi as Casanova. Television Casanova, a 1971 BBC Television serial, written by Dennis Potter and starring Frank Finlay Casanova, a 2005 BBC Television serial featuring David Tennant as young Casanova and Peter O'Toole as the older Casanova In 2017, an episode of Horrible Histories called "Ridiculous Romantics" featured Tom Stourton, portraying Casanova.
Written works Casanovas Heimfahrt (Casanova's Homecoming) (1918) by Arthur Schnitzler The Venetian Glass Nephew (1925) by Elinor Wylie, in which Casanova appears as a major character under the transparent pseudonym "Chevalier de Chastelneuf" Széljegyzetek Casanovához (Marginalia on Casanova) (1939) by Miklós Szentkuthy Vendégjáték Bolzanóban (Conversations in Bolzano or Casanova in Bolzano) (1940), a novel by Sándor Márai Le Bonheur ou le Pouvoir (1980), by Pierre Kast The Fortunes of Casanova and Other Stories (1994), by Rafael Sabatini, includes nine stories (originally published 1914–1921) based on incidents in Casanova's memoirs Casanova (1998), a novel by Andrew Miller Casanova, Dernier Amour (2000), by Pascal Lainé Casanova in Bohemia (2002), a novel about Casanova's last years at Dux, Bohemia, by Andrei Codrescu Een Schitterend Gebrek (English title In Lucia's Eyes), a 2003 Dutch novel by Arthur Japin, in which Casanova's youthful amour Lucia is viewed as the love of his life "A Disciple of Plato", a short story by English writer Robert Aickman, first printed in the 2015 posthumous collection The Strangers and Other Writings, in which the main character—throughout described as "the philosopher"—is revealed in the last lines to be Casanova.
See also Manon Balletti Don Juan Notes and references Bibliography External links Memoirs of Jacques Casanova de Seingalt 1725–1798 Ebook Category:1725 births Category:1798 deaths Category:18th-century Italian writers Category:18th-century male writers Category:18th-century Italian novelists Category:Duellists Category:Italian escapees Category:Italian Freemasons Category:Italian librarians Category:Italian memoirists Category:Italian Roman Catholics Category:Italian writers in French Category:People from Venice Category:Translators of Homer Category:Articles containing unlinked shortened footnotes
In continuum mechanics, the Cauchy stress tensor , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the traction vector T(n) across an imaginary surface perpendicular to n: where, The SI units of both stress tensor and stress vector are N/m2, corresponding to the stress scalar. The unit vector is dimensionless. The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates.
A graphical representation of this transformation law is the Mohr's circle for stress. The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: It is a central concept in the linear theory of elasticity. For large deformations, also called finite deformations, other measures of stress are required, such as the Piola–Kirchhoff stress tensor, the Biot stress tensor, and the Kirchhoff stress tensor. According to the principle of conservation of linear momentum, if the continuum body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations (Cauchy's equations of motion for zero acceleration).
At the same time, according to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components, instead of the original nine. However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric. This also is the case when the Knudsen number is close to one, , or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers. There are certain invariants associated with the stress tensor, whose values do not depend upon the coordinate system chosen, or the area element upon which the stress tensor operates.
These are the three eigenvalues of the stress tensor, which are called the principal stresses. Euler–Cauchy stress principle – stress vector The Euler–Cauchy stress principle states that upon any surface (real or imaginary) that divides the body, the action of one part of the body on the other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body, and it is represented by a field , called the traction vector, defined on the surface and assumed to depend continuously on the surface's unit vector . To formulate the Euler–Cauchy stress principle, consider an imaginary surface passing through an internal material point dividing the continuous body into two segments, as seen in Figure 2.1a or 2.1b (one may use either the cutting plane diagram or the diagram with the arbitrary volume inside the continuum enclosed by the surface ).
Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces and body forces . Thus, the total force applied to a body or to a portion of the body can be expressed as: Only surface forces will be discussed in this article as they are relevant to the Cauchy stress tensor. When the body is subjected to external surface forces or contact forces , following Euler's equations of motion, internal contact forces and moments are transmitted from point to point in the body, and from one segment to the other through the dividing surface , due to the mechanical contact of one portion of the continuum onto the other (Figure 2.1a and 2.1b).
On an element of area containing , with normal vector , the force distribution is equipollent to a contact force exerted at point P and surface moment . In particular, the contact force is given by where is the mean surface traction. Cauchy's stress principle asserts that as becomes very small and tends to zero the ratio becomes and the couple stress vector vanishes. In specific fields of continuum mechanics the couple stress is assumed not to vanish; however, classical branches of continuum mechanics address non-polar materials which do not consider couple stresses and body moments. The resultant vector is defined as the surface traction, also called stress vector, traction, or traction vector.
given by at the point associated with a plane with a normal vector : This equation means that the stress vector depends on its location in the body and the orientation of the plane on which it is acting. This implies that the balancing action of internal contact forces generates a contact force density or Cauchy traction field that represents a distribution of internal contact forces throughout the volume of the body in a particular configuration of the body at a given time . It is not a vector field because it depends not only on the position of a particular material point, but also on the local orientation of the surface element as defined by its normal vector .
Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, i.e. parallel to , and can be resolved into two components (Figure 2.1c): one normal to the plane, called normal stress where is the normal component of the force to the differential area and the other parallel to this plane, called the shear stress where is the tangential component of the force to the differential surface area . The shear stress can be further decomposed into two mutually perpendicular vectors. Cauchy’s postulate According to the Cauchy Postulate, the stress vector remains unchanged for all surfaces passing through the point and having the same normal vector at , i.e., having a common tangent at .
This means that the stress vector is a function of the normal vector only, and is not influenced by the curvature of the internal surfaces. Cauchy’s fundamental lemma A consequence of Cauchy's postulate is Cauchy’s Fundamental Lemma, also called the Cauchy reciprocal theorem, which states that the stress vectors acting on opposite sides of the same surface are equal in magnitude and opposite in direction. Cauchy's fundamental lemma is equivalent to Newton's third law of motion of action and reaction, and is expressed as Cauchy’s stress theorem—stress tensor The state of stress at a point in the body is then defined by all the stress vectors T(n) associated with all planes (infinite in number) that pass through that point.
However, according to Cauchy’s fundamental theorem, also called Cauchy’s stress theorem, merely by knowing the stress vectors on three mutually perpendicular planes, the stress vector on any other plane passing through that point can be found through coordinate transformation equations. Cauchy's stress theorem states that there exists a second-order tensor field σ(x, t), called the Cauchy stress tensor, independent of n, such that T is a linear function of n: This equation implies that the stress vector T(n) at any point P in a continuum associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e.
in terms of the components σij of the stress tensor σ. To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area dA oriented in an arbitrary direction specified by a normal unit vector n (Figure 2.2). The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane with unit normal n. The stress vector on this plane is denoted by T(n). The stress vectors acting on the faces of the tetrahedron are denoted as T(e1), T(e2), and T(e3), and are by definition the components σij of the stress tensor σ.
This tetrahedron is sometimes called the Cauchy tetrahedron. The equilibrium of forces, i.e. Euler's first law of motion (Newton's second law of motion), gives: where the right-hand-side represents the product of the mass enclosed by the tetrahedron and its acceleration: ρ is the density, a is the acceleration, and h is the height of the tetrahedron, considering the plane n as the base. The area of the faces of the tetrahedron perpendicular to the axes can be found by projecting dA into each face (using the dot product): and then substituting into the equation to cancel out dA: To consider the limiting case as the tetrahedron shrinks to a point, h must go to 0 (intuitively, the plane n is translated along n toward O).
As a result, the right-hand-side of the equation approaches 0, so Assuming a material element (Figure 2.3) with planes perpendicular to the coordinate axes of a Cartesian coordinate system, the stress vectors associated with each of the element planes, i.e. T(e1), T(e2), and T(e3) can be decomposed into a normal component and two shear components, i.e. components in the direction of the three coordinate axes.
For the particular case of a surface with normal unit vector oriented in the direction of the x1-axis, denote the normal stress by σ11, and the two shear stresses as σ12 and σ13: In index notation this is The nine components σij of the stress vectors are the components of a second-order Cartesian tensor called the Cauchy stress tensor, which completely defines the state of stress at a point and is given by where σ11, σ22, and σ33 are normal stresses, and σ12, σ13, σ21, σ23, σ31, and σ32 are shear stresses. The first index i indicates that the stress acts on a plane normal to the Xi -axis, and the second index j denotes the direction in which the stress acts (For example, σ12 implies that the stress is acting on the plane that is normal to the 1st axis i.e.
;X1 and acts along the 2nd axis i.e.;X2). A stress component is positive if it acts in the positive direction of the coordinate axes, and if the plane where it acts has an outward normal vector pointing in the positive coordinate direction. Thus, using the components of the stress tensor or, equivalently, Alternatively, in matrix form we have The Voigt notation representation of the Cauchy stress tensor takes advantage of the symmetry of the stress tensor to express the stress as a six-dimensional vector of the form: The Voigt notation is used extensively in representing stress–strain relations in solid mechanics and for computational efficiency in numerical structural mechanics software.
Transformation rule of the stress tensor It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an xi-system to an xi' -system, the components σij in the initial system are transformed into the components σij' in the new system according to the tensor transformation rule (Figure 2.4): where A is a rotation matrix with components aij. In matrix form this is Expanding the matrix operation, and simplifying terms using the symmetry of the stress tensor, gives The Mohr circle for stress is a graphical representation of this transformation of stresses.
Normal and shear stresses The magnitude of the normal stress component σn of any stress vector T(n) acting on an arbitrary plane with normal unit vector n at a given point, in terms of the components σij of the stress tensor σ, is the dot product of the stress vector and the normal unit vector: The magnitude of the shear stress component τn, acting orthogonal to the vector n, can then be found using the Pythagorean theorem: where Balance laws – Cauchy's equations of motion Cauchy's first law of motion According to the principle of conservation of linear momentum, if the continuum body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations.
For example, for a hydrostatic fluid in equilibrium conditions, the stress tensor takes on the form: where is the hydrostatic pressure, and is the kronecker delta. {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations |- |Consider a continuum body (see Figure 4) occupying a volume , having a surface area , with defined traction or surface forces per unit area acting on every point of the body surface, and body forces per unit of volume on every point within the volume .
Thus, if the body is in equilibrium the resultant force acting on the volume is zero, thus: By definition the stress vector is , then Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives For an arbitrary volume the integral vanishes, and we have the equilibrium equations |} Cauchy's second law of motion According to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components, instead of the original nine: {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of symmetry of the stress tensor |- | Summing moments about point O (Figure 4) the resultant moment is zero as the body is in equilibrium.
Thus, where is the position vector and is expressed as Knowing that and using Gauss's divergence theorem to change from a surface integral to a volume integral, we have The second integral is zero as it contains the equilibrium equations. This leaves the first integral, where , therefore For an arbitrary volume V, we then have which is satisfied at every point within the body. Expanding this equation we have , , and or in general This proves that the stress tensor is symmetric |} However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric.
This also is the case when the Knudsen number is close to one, , or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers. Principal stresses and stress invariants At every point in a stressed body there are at least three planes, called principal planes, with normal vectors , called principal directions, where the corresponding stress vector is perpendicular to the plane, i.e., parallel or in the same direction as the normal vector , and where there are no normal shear stresses . The three stresses normal to these principal planes are called principal stresses.
The components of the stress tensor depend on the orientation of the coordinate system at the point under consideration. However, the stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it. There are certain invariants associated with every tensor which are also independent of the coordinate system. For example, a vector is a simple tensor of rank one. In three dimensions, it has three components. The value of these components will depend on the coordinate system chosen to represent the vector, but the magnitude of the vector is a physical quantity (a scalar) and is independent of the Cartesian coordinate system chosen to represent the vector (so long as it is normal).
Similarly, every second rank tensor (such as the stress and the strain tensors) has three independent invariant quantities associated with it. One set of such invariants are the principal stresses of the stress tensor, which are just the eigenvalues of the stress tensor. Their direction vectors are the principal directions or eigenvectors. A stress vector parallel to the normal unit vector is given by: where is a constant of proportionality, and in this particular case corresponds to the magnitudes of the normal stress vectors or principal stresses. Knowing that and , we have This is a homogeneous system, i.e. equal to zero, of three linear equations where are the unknowns.
To obtain a nontrivial (non-zero) solution for , the determinant matrix of the coefficients must be equal to zero, i.e. the system is singular. Thus, Expanding the determinant leads to the characteristic equation where The characteristic equation has three real roots , i.e. not imaginary due to the symmetry of the stress tensor. The , and , are the principal stresses, functions of the eigenvalues . The eigenvalues are the roots of the characteristic polynomial. The principal stresses are unique for a given stress tensor. Therefore, from the characteristic equation, the coefficients , and , called the first, second, and third stress invariants, respectively, always have the same value regardless of the coordinate system's orientation.
For each eigenvalue, there is a non-trivial solution for in the equation . These solutions are the principal directions or eigenvectors defining the plane where the principal stresses act. The principal stresses and principal directions characterize the stress at a point and are independent of the orientation. A coordinate system with axes oriented to the principal directions implies that the normal stresses are the principal stresses and the stress tensor is represented by a diagonal matrix: The principal stresses can be combined to form the stress invariants, , , and . The first and third invariant are the trace and determinant respectively, of the stress tensor.
Thus, Because of its simplicity, the principal coordinate system is often useful when considering the state of the elastic medium at a particular point. Principal stresses are often expressed in the following equation for evaluating stresses in the x and y directions or axial and bending stresses on a part. The principal normal stresses can then be used to calculate the von Mises stress and ultimately the safety factor and margin of safety. Using just the part of the equation under the square root is equal to the maximum and minimum shear stress for plus and minus. This is shown as: Maximum and minimum shear stresses The maximum shear stress or maximum principal shear stress is equal to one-half the difference between the largest and smallest principal stresses, and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses, i.e.
the plane of the maximum shear stress is oriented from the principal stress planes. The maximum shear stress is expressed as Assuming then When the stress tensor is non zero the normal stress component acting on the plane for the maximum shear stress is non-zero and it is equal to {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of the maximum and minimum shear stresses |- |The normal stress can be written in terms of principal stresses as Knowing that , the shear stress in terms of principal stresses components is expressed as The maximum shear stress at a point in a continuum body is determined by maximizing subject to the condition that This is a constrained maximization problem, which can be solved using the Lagrangian multiplier technique to convert the problem into an unconstrained optimization problem.
Thus, the stationary values (maximum and minimum values)of occur where the gradient of is parallel to the gradient of . The Lagrangian function for this problem can be written as where is the Lagrangian multiplier (which is different from the use to denote eigenvalues). The extreme values of these functions are thence These three equations together with the condition may be solved for and By multiplying the first three equations by and , respectively, and knowing that we obtain Adding these three equations we get this result can be substituted into each of the first three equations to obtain Doing the same for the other two equations we have A first approach to solve these last three equations is to consider the trivial solution .
However, this option does not fulfill the constraint . Considering the solution where and , it is determine from the condition that , then from the original equation for it is seen that . The other two possible values for can be obtained similarly by assuming and and Thus, one set of solutions for these four equations is: These correspond to minimum values for and verifies that there are no shear stresses on planes normal to the principal directions of stress, as shown previously. A second set of solutions is obtained by assuming and .
Thus we have To find the values for and we first add these two equations Knowing that for and we have and solving for we have Then solving for we have and The other two possible values for can be obtained similarly by assuming and and Therefore, the second set of solutions for , representing a maximum for is Therefore, assuming , the maximum shear stress is expressed by and it can be stated as being equal to one-half the difference between the largest and smallest principal stresses, acting on the plane that bisects the angle between the directions of the largest and smallest principal stresses.
|} Stress deviator tensor The stress tensor can be expressed as the sum of two other stress tensors: a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor, , which tends to change the volume of the stressed body; and a deviatoric component called the stress deviator tensor, , which tends to distort it. So: where is the mean stress given by Pressure () is generally defined as negative one-third the trace of the stress tensor minus any stress the divergence of the velocity contributes with, i.e. where is a proportionality constant, is the divergence operator, is the k:th Cartesian coordinate, is the velocity and is the k:th Cartesian component of .
The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the Cauchy stress tensor: Invariants of the stress deviator tensor As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor are the same as the principal directions of the stress tensor . Thus, the characteristic equation is where , and are the first, second, and third deviatoric stress invariants, respectively.
Their values are the same (invariant) regardless of the orientation of the coordinate system chosen. These deviatoric stress invariants can be expressed as a function of the components of or its principal values , , and , or alternatively, as a function of or its principal values , , and . Thus, Because , the stress deviator tensor is in a state of pure shear. A quantity called the equivalent stress or von Mises stress is commonly used in solid mechanics. The equivalent stress is defined as Octahedral stresses Considering the principal directions as the coordinate axes, a plane whose normal vector makes equal angles with each of the principal axes (i.e.
having direction cosines equal to ) is called an octahedral plane. There are a total of eight octahedral planes (Figure 6). The normal and shear components of the stress tensor on these planes are called octahedral normal stress and octahedral shear stress , respectively. Octahedral plane passing through the origin is known as the π-plane (π not to be confused with mean stress denoted by π in above section) . On the π-plane, . Knowing that the stress tensor of point O (Figure 6) in the principal axes is the stress vector on an octahedral plane is then given by: The normal component of the stress vector at point O associated with the octahedral plane is which is the mean normal stress or hydrostatic stress.
This value is the same in all eight octahedral planes. The shear stress on the octahedral plane is then See also Critical plane analysis References Category:Tensors Category:Solid mechanics Category:Continuum mechanics
2016 in film is an overview of events, including the highest-grossing films, award ceremonies, festivals, and a list of films released and deaths. Evaluation of the year In his article highlighting the best films of 2016, Richard Brody of The New Yorker stated, "Hollywood is the world’s best money-laundering machine. It takes in huge amounts of money from the sale of mass-market commodities and cleanses some of it with the production of cinematic masterworks. Earning billions of dollars from C.G.I. comedies for children, superhero movies, sci-fi apocalypses, and other popular genres, the big studios channel some of those funds into movies by Wes Anderson, Sofia Coppola, Spike Lee, Martin Scorsese, James Gray, and other worthies.
Sometimes there’s even an overlap between the two groups of movies, as when Ryan Coogler made Creed, or when Scorsese made the modernist horror instant-classic Shutter Island, or when Clint Eastwood makes just about anything." Highest-grossing films The top ten films released in 2016 by worldwide gross are as follows: Captain America: Civil War, Rogue One: A Star Wars Story, Finding Dory, and Zootopia grossed more than $1 billion each, making them among the highest-grossing films of all time. This is the first year that two animated films (Finding Dory and Zootopia) grossed over $1 billion in a single year and are ranked the 8th and 9th highest-grossing animated films respectively.
The Jungle Book is among the 50 highest-grossing films of all time. The Secret Life of Pets is the 17th-highest-grossing animated film of all time. Captain America: Civil War, Zootopia, Kung Fu Panda 3, Warcraft, and The Great Wall have all grossed more than ¥1 billion at the Chinese box office, making them among the highest-grossing films in China. 2016 box office records Studio records Walt Disney Studios reached $1 billion at the domestic box office faster than any other studio; it reached this goal on the 128th day of 2016, beating Universal Studios' record of reaching the goal on the 165th day of 2015.
Disney's previous record for reaching $1 billion was on the 174th day of 2015. The studio became the first to have five of its releases (Rogue One, Finding Dory, Captain America: Civil War, The Jungle Book, and Zootopia) from a single year reach $300 million domestically. Disney also eclipsed Universal's 2015 record for most films from a single year crossing $1 billion worldwide with four (Captain America: Civil War, Rogue One, Finding Dory, and Zootopia), setting a new record for most billion-dollar-grossing films over two years with six (including Age of Ultron and The Force Awakens). Walt Disney Studios has also become the first studio to have the five highest-grossing films worldwide, and the first since at least 1913 to have the three highest-grossing films in the U.S., both in a single year.
Disney became the first studio to gross more than $3 billion at the domestic box office and, with the release of Rogue One, became the first to gross more than $7 billion at the global box office, surpassing Universal's previous record of $6.9 billion in 2015. Disney later passes $7 billion at the global box office again in 2018. Disney is also the first studio to have three films gross over $400 million domestically in a single year (Rogue One, Finding Dory, and Captain America: Civil War), and the first to fill in all slots of the top five films of any particular year.