Formulação dual do método dos elementos de contorno anisotrópico

❋♦r♠✉❧❛çã♦ ❉✉❛❧ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦

❆♥✐s♦tró♣✐❝♦

▲❯■❩ ❍❊◆❘■◗❯❊ ❉❆ ❙■▲❱❆ ❋❊❘◆➪◆❉❊❩

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛

❛♦

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❞❡ ❊str✉t✉r❛s

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

▼❡str❡ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧

Pr♦❣r❛♠❛✿ ❊♥❣❡♥❤❛r✐❛ ❞❡ ❊str✉t✉r❛s

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❚✐t✳ ❲✐❧s♦♥ ❙❡r❣✐♦ ❱❡♥t✉r✐♥✐

❖r✐❡♥t❛❞♦r ❞❡s✐❣♥❛❞♦✿ Pr♦❢✳ ❉r✳ ❏♦ã♦ ❇❛t✐st❛ ❞❡ P❛✐✈❛

❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❆P❊❙

AUTORIZO A REPRODUÇÃO TOTAL OU PARCIAL DESTE TRABALHO, POR QUALQUER MEIO CONVENCIONAL OU ELETRÔNICO, PARA FINS DE ESTUDO E PESQUISA, DESDE QUE CITADA A FONTE.

Ficha catalográfica preparada pela Seção de Atendimentos ao Usuário do Serviço de Biblioteca – EESC/USP.

Fernández, Luiz Henrique da Silva

F363f Formulação dual do método dos elementos de contorno anisotrópico. / Luiz Henrique da Silva Fernández; orientador João Batista de Paiva. São Carlos, 2012.

Dissertação (Mestrado) - Programa de Pós-Graduação em Engenharia de Estruturas -- Escola de Engenharia de São Carlos da Universidade de São Paulo, 2012.

❋♦r♠✉❧❛çã♦ ❉✉❛❧ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦

❆♥✐s♦tró♣✐❝♦

❊st❛ ❞✐ss❡rt❛çã♦ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ r❡❛❧✐③❛❞❛ ♣♦r ▲✉✐③ ❍❡♥r✐q✉❡ ❞❛ ❙✐❧✈❛ ❋❡r♥á♥❞❡③ ❡♠ ✷✼✴✵✼✴✷✵✶✷✳ ❖ ♦r✐❣✐♥❛❧ ❡♥❝♦♥tr❛✲s❡ ❞✐s♣♦♥í✈❡❧ ♥♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❞❡ ❊str✉t✉r❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ♠✐♥❤❛ q✉❡r✐❞❛ ▲❡tí❝✐❛ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡ ❝❛r✐♥❤♦✳ ∞

❆♦s ♠❡✉s ♣❛✐s✱ à ♠✐♥❤❛ t✐❛ ❚❡r❡③❛ ❈r✐st✐♥❛ ▲♦♣❡③ ❋❡r♥á♥❞❡③ ❡ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ❛✈ó ➪✉r❡❛ ▲♦♣❡③ ❋❡r♥á♥❞❡③ ❝✉❥♦ ♦ ❛♠♦r ♣❡❧❛ ❡♥❣❡♥❤❛r✐❛ ♠❡ ❢❛s❝✐♥❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ▼❛t❡✉s ◆♦❣✉❡✐r❛✱ ▼❛r❦✉s ❘❡❜♠❛♥♥✱ ❍✉♠❜❡rt♦ ❇♦r❛tt♦ ❡ ●❛❜r✐❡❧ ❙❡♥❛r❡③✐ ♣❡❧♦ ❛♣♦✐♦✳

❆♦s ♠❡✉s ♦r✐❡♥t❛❞♦r❡s✱ Pr♦❢✳ ❲✐❧s♦♥ ❱❡♥t✉r✐♥✐ ❡ Pr♦❢✳ ❏♦ã♦ P❛✐✈❛✱ ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦✳ ❆♦s ♣r♦❢❡ss♦r❡s✱ ❡ ❛♠✐❣♦s✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ Pr♦❢✳ ❊st❡✈ã♦ ❇✐❝❛❧❤♦✱ Pr♦❢✳ ❆r♠❛♥❞♦ ▲❛✈❛❧❧✱ Pr♦❢✳ ❘✐❝❛r❞♦ ❋❛❦✉r② ❡ Pr♦❢✳ ❘♦q✉❡ P✐t❛♥❣✉❡✐r❛✳ ❆❝r❡❞✐t♦ q✉❡ ❝❛❞❛ ✉♠ ❞❡❧❡s s❛✐❜❛ ❞❡ q✉❡ ❢♦r♠❛ í♠♣❛r ❝♦♥tr✐❜✉ír❛♠ ♣r❛ ❡st❡ ♠♦♠❡♥t♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❲✐❧s♦♥ ❱❡♥t✉r✐♥✐✱ ❍✉♠❜❡rt♦ ❈♦❞❛ ❡ ❙❛♠✉❡❧ ●✐♦♥❣♦ q✉❡ t♦r♥❛r❛♠ ❡st❛ ❡①♣❡r✐ê♥❝✐❛ ❛✐♥❞❛ ♠❛✐s ✐♥t❡r❡ss❛♥t❡✳

❆♦s ❝♦❧❡❣❛s ❞❡ ♠❡str❛❞♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❆r❡❢ ❑③❛♠✱ ❛ ❙♦❝♦rr♦ ❙❛♠♣❛✐♦ ❡ ❛ ❊✉♥✐❝❡ ❙❛♥t♦s ♣❡❧❛ ❝♦♠♣❛♥❤✐❛✳

❆ t♦❞♦s ♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❊str✉t✉r❛s ❞❛ ❯❙P q✉❡✱ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

➚ ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❝♦♥❝❡❞✐❞♦✳

❘❡s✉♠♦

❊st❡ t❡①t♦ tr❛t❛ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧ ❡♠♣r❡❣❛♥❞♦ ❛ s♦❧✉çã♦ ❢✉♥❞❛✲ ♠❡♥t❛❧ ❛♥✐s♦tró♣✐❝❛✳ ❆s ✐♥t❡❣r❛✐s ✐♠♣ró♣r✐❛s q✉❡ s✉r❣❡♠ ♥❡st❛ ❢♦r♠✉❧❛çã♦ sã♦ r❡❣✉❧❛r✐③❛❞❛s ♣❡❧❛ té❝♥✐❝❛ ❞❛ s✉❜tr❛çã♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡✳ ❆♣❧✐❝❛✲s❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛✉t♦✲❛❞❛♣t❛t✐✈❛ ❞❡ ❚❡❧❧❡s ♣❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞❛s ✐♥t❡❣r❛✐s q✉❛s❡✲s✐♥❣✉❧❛r❡s✳ ❆♣r❡s❡♥t❛✲s❡ ♦ ♣r♦❣r❛♠❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❡s❡♥✈♦❧✈✐❞♦ ✉t✐❧✐③❛♥❞♦ ♦s ♣❛r❛❞✐❣♠❛s ❞❛ ♣r♦❣r❛♠❛çã♦ ♦r✐❡♥t❛❞❛ ❛ ♦❜❥❡t♦s ❡ ♣r♦❝❡ss❛♠❡♥t♦ ❡♠ ♣❛r❛❧❡❧♦✳ ❋♦r❛♠ ❛♥❛❧✐s❛❞♦s ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠♣❛r❛❞♦s àq✉❡❧❡s ❞❛ s♦❧✉✲ çã♦ ❛♥❛❧ít✐❝❛✳ ❖s r❡s✉❧t❛❞♦s ❛❧❝❛♥ç❛❞♦s ♠♦str❛r❛♠✲s❡ s❛t✐s❢❛tór✐♦s ✈❛❧✐❞❛♥❞♦ ❛ ❢♦r♠✉❧❛çã♦ ♣r♦♣♦st❛✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧✱ ❆♥✐s♦tr♦♣✐❛✱ ❘❡❣✉❧❛r✐③❛çã♦ ❞❡ ■♥t❡❣r❛✐s ■♠♣ró♣r✐❛s✱ Pr♦❝❡ss❛♠❡♥t♦ ❡♠ P❛r❛❧❡❧♦✳

❆❜str❛❝t

❚❤✐s t❡①t ❞❡❛❧s ✇✐t❤ t❤❡ ❉✉❛❧ ❇♦✉♥❞❛r② ❊❧❡♠❡♥t ❋♦r♠✉❧❛t✐♦♥ ▼❡t❤♦❞ ✉s✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥ ❢♦r ❛♥✐s♦tr♦♣✐❝ ❜♦❞②✳ ❚❤❡ ✐♠♣r♦♣❡r ✐♥t❡❣r❛❧s t❤❛t ❛r✐s❡ ✐♥ t❤✐s ❢♦r♠✉❧❛t✐♦♥ ❛r❡ r❡❣✉❧❛r✐③❡❞ ✉s✐♥❣ t❤❡ s✐♥❣✉❧❛r✐t② s✉❜tr❛❝t✐♦♥ t❡❝❤♥✐q✉❡✳ ❚❤❡ s❡❧❢✲❛❞❛♣t✐✈❡ ❝♦♦r❞✐♥❛t❡ tr❛♥s❢♦r♠❛t✐♦♥ ❞❡✈❡❧♦♣❡❞ ❜② ❚❡❧❧❡s ✐s ✉s❡❞ t♦ ❡✈❛❧✉❛t❡ t❤❡ ♥❡❛r✲s✐♥❣✉❧❛r ✐♥t❡❣r❛❧s✳ ❚❤❡ ❝♦♠♣✉t❡r ♣r♦❣r❛♠ ❞❡✈❡❧♦♣❡❞ ✉s✐♥❣ t❤❡ ♣❛r❛❞✐❣♠s ♦❢ ♦❜❥❡❝t✲♦r✐❡♥t❡❞ ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ♣❛r❛❧❧❡❧ ♣r♦❝❡ss✐♥❣ ✐s ♣r❡s❡♥t❡❞✳ ❙❡✈❡r❛❧ ♣r♦❜❧❡♠s ✇❡r❡ ❛♥❛❧②③❡❞ ❛♥❞ ✐ts r❡s✉❧ts ❝♦♠♣❛r❡❞ ✇✐t❤ t❤♦s❡ ♣r♦♣♦s❡❞ ❜② ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥✳ ❚❤❡ r❡s✉❧ts ❛❝❤✐❡✈❡❞ ✇❡r❡ s❛t✐s❢❛❝t♦r② t❤❡r❡❢♦r❡ ✈❛❧✐❞❛t✐♥❣ t❤❡ ♣r♦♣♦s❡❞ ❢♦r♠✉❧❛t✐♦♥✳

❑❡②✇♦r❞s✿ ❉✉❛❧ ❇♦✉♥❞❛r② ❊❧❡♠❡♥t ▼❡t❤♦❞✱ ❆♥✐s♦tr♦♣②✱ ❘❡❣✉❧❛r✐③❛t✐♦♥ ♦❢ ■♠♣r♦♣❡r ■♥t❡❣r❛❧s✱ P❛r❛❧❧❡❧ Pr♦❝❡ss✐♥❣✳

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ①✐

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐✐✐

▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✈

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❈♦♥tr✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ ✺

✸ ❚❡♦r✐❛ ❞❛ ❊❧❛st✐❝✐❞❛❞❡ ✾

✸✳✶ ❉❡❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✸✳✷ ❚❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✸ ❊q✉❛çõ❡s ❞❡ ❊q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✹ ❊q✉❛çõ❡s ❞❡ ❈♦♠♣❛t✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✺ ▲❡✐ ❞❡ ❍♦♦❦❡ ●❡♥❡r❛❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✻ ❊st❛❞♦ P❧❛♥♦ ❞❡ ❉❡❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✼ ❊st❛❞♦ P❧❛♥♦ ❞❡ ❚❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✽ ❙♦❧✉çã♦ ❋✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹ ❊q✉❛çõ❡s ■♥t❡❣r❛✐s ❞❡ ❈♦♥t♦r♥♦ ✷✼

✹✳✶ ▼ét♦❞♦ ❞♦s ❘❡sí❞✉♦s P♦♥❞❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✷ ❊q✉❛çõ❡s ■♥t❡❣r❛✐s ❞❡ ❈♦♥t♦r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✺ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧ ✸✸

✺✳✶ ❊❧❡♠❡♥t♦s ■s♦♣❛r❛♠étr✐❝♦s ❞❡ ❈♦♥t♦r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✷ ▼ét♦❞♦ ❞❛ ❙✉❜tr❛çã♦ ❞❡ ❙✐♥❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✷✳✶ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❞♦ ❚✐♣♦ln_|r_| ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✺✳✷✳✷ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❞♦ ❚✐♣♦|r_|−1 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵} ✺✳✷✳✸ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❞♦ ❚✐♣♦|r_|−2 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶} ✺✳✸ ■♥t❡❣r❛çã♦ ◆✉♠ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✹ ❙✐st❡♠❛ ❞❡ ❊q✉❛çõ❡s ❆❧❣é❜r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

① ❙❯▼➪❘■❖

✻ ■♠♣❧❡♠❡♥t❛çã♦ ❈♦♠♣✉t❛❝✐♦♥❛❧ ✹✾

✻✳✶ Pr♦❣r❛♠❛çã♦ ❖r✐❡♥t❛❞❛ ❛ ❖❜❥❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✷ ❏❛✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✻✳✸ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✻✳✸✳✶ ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✻✳✸✳✷ ❆ss❡♠❜❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✸✳✸ ❙♦❧✈❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✼ ❊①❡♠♣❧♦s ✺✼

✼✳✶ ❈❤❛♣❛ s♦❜ ❚❡♥sã♦ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✼✳✷ ❈❤❛♣❛ s♦❜ ❈✐s❛❧❤❛♠❡♥t♦ P✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✼✳✸ ❱✐❣❛ s♦❜ ❋❧❡①ã♦ P✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✼✳✹ ❱✐❣❛ ❊♥❣❛st❛❞❛ s♦❜ ❈❛rr❡❣❛♠❡♥t♦ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✼✳✺ ❋✉r♦ ❊❧í♣t✐❝♦ ❡♠ ▼❡✐♦ ■♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✼✳✻ ❋r❛t✉r❛ ❡♠ ▼❡✐♦ ■♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾

✽ ❈♦♥❝❧✉sõ❡s ✼✼

✽✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✽✳✷ ❙✉❣❡stõ❡s ♣❛r❛ P❡sq✉✐s❛s ❋✉t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

❆ ❉❡❧t❛ ❞❡ ❉✐r❛❝ ✼✾

❇ ❙✉❜tr❛çã♦ ❞❡ ❙✐♥❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ❊❧❡♠❡♥t♦ ▲✐♥❡❛r ✽✶

❈ ❚❡♥sõ❡s ❙♦❜r❡ ♦ ❈♦♥t♦r♥♦ ✽✺

❉ ❊rr♦ ❆❜s♦❧✉t♦ ❡ ❘❡❧❛t✐✈♦ ✽✾

❊ ❉❛❞♦s ❞♦ ❚❡♠♣♦ ❞❡ Pr♦❝❡ss❛♠❡♥t♦ ✾✶

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

c ▼❛tr✐③ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❡❧ást✐❝♦s ❞❡ ✢❡①✐❜✐❧✐❞❛❞❡ n ❱❡t♦r ♥♦r♠❛❧

t ❋♦rç❛ ❞❡ s✉♣❡r❢í❝✐❡ u ❉❡s❧♦❝❛♠❡♥t♦

z ❈♦♦r❞❡♥❛❞❛ ❝♦♠♣❧❡①❛ ❞♦ ♣♦♥t♦ ❝❛♠♣♦ z′ ❈♦♦r❞❡♥❛❞❛ ❝♦♠♣❧❡①❛ ❞♦ ♣♦♥t♦ ❢♦♥t❡

C ▼❛tr✐③ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❡❧ást✐❝♦s ❞❡ r✐❣✐❞❡③ E ▼ó❞✉❧♦ ❞❡ ❞❡❢♦r♠❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ G ▼ó❞✉❧♦ ❞❡ ❞❡❢♦r♠❛çã♦ tr❛♥s✈❡rs❛❧ J ❏❛❝♦❜✐❛♥♦

δ ❉❡❧t❛ ❞❡ ❉✐r❛❝ ε ❉❡❢♦r♠❛çã♦

η ❈♦❡✜❝✐❡♥t❡s ❞❡ ✐♥✢✉ê♥❝✐❛ ♠út✉❛ ❞❡ ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ t✐♣♦ µ ❘❛✐③ ❞❛ ❡q✉❛çã♦ ❝❛r❛❝t❡ríst✐❝❛

ν ❈♦❡✜❝✐❡♥t❡ ❞❡ P♦✐ss♦♥ ξ ❈♦♦r❞❡♥❛❞❛ ❛❞✐♠❡♥s✐♦♥❛❧ ρ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❈❤❡♥ts♦✈ σ ❚❡♥sã♦

φ ❋✉♥çã♦ ❞❡ ❢♦r♠❛

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✸✳✶ ❱❡t♦r ❞❡s❧♦❝❛♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✷ ❉❡❢♦r♠❛çã♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✸ ❋♦rç❛ ❞❡ s✉♣❡r❢í❝✐❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✹ ❈♦♠♣♦♥❡♥t❡s ❞❡ t❡♥sã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✺ ❊q✉✐❧í❜r✐♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✻ ❊q✉✐❧í❜r✐♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ♣r✐s♠át✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✼ ❉✐❛❣r❛♠❛ t❡♥sã♦✲❞❡❢♦r♠❛çã♦ tí♣✐❝❛ ❞♦ ❛ç♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✽ ❊st❛❞♦ ♣❧❛♥♦ ❞❡ ❞❡❢♦r♠❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✾ ❊st❛❞♦ ♣❧❛♥♦ ❞❡ t❡♥sã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶✵ ▼❛♣❡❛♠❡♥t♦ ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✶✶ ❙♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹✳✶ P♦♥t♦ ❢♦♥t❡ s♦❜r❡ ♦ ❝♦♥t♦r♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✺✳✶ ▼ét♦❞♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ❝♦♥t♦r♥♦ ❞✉❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✷ ❉✐s❝r❡t✐③❛çã♦ ❞♦ ❝♦♥t♦r♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✸ P❛r❛♠❡tr✐③❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ✉♠ ❡❧❡♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✹ ❊❢❡✐t♦ ❘✉♥❣❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✺✳✺ ❏❛❝♦❜✐❛♥♦ ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✻ ❘❡❣✉❧❛r✐③❛çã♦ ❞♦ ✐♥t❡❣r❛♥❞♦ s✐♥❣✉❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✼ ❚r❛♥s❢♦r♠❛çã♦ ❝ú❜✐❝❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✽ ❊①❡♠♣❧♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡❣r❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✾ ●❡r❛çã♦ ❞❛s ♠❛tr✐③❡s ❞❡ ✐♥✢✉ê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✻✳✶ ❊str✉t✉r❛ ♣r✐♥❝✐♣❛❧ ❞♦ ♣r♦❣r❛♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✻✳✷ ❊str✉t✉r❛ ❞♦ ▼♦❞❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✻✳✸ ❊str✉t✉r❛ ❞♦ ❆ss❡♠❜❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✼✳✶ ❈❤❛♣❛ s♦❜ t❡♥sã♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✼✳✷ ❉✐s❝r❡t✐③❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❝❤❛♣❛ s♦❜ t❡♥sã♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✼✳✸ ❈♦♥✜❣✉r❛çã♦ ❞❡❢♦r♠❛❞❛ ❞♦ ♣r♦❜❧❡♠❛ ❝❤❛♣❛ s♦❜ t❡♥sã♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✼✳✹ ❈❤❛♣❛ s♦❜ ❝✐s❛❧❤❛♠❡♥t♦ ♣✉r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✼✳✺ ❈♦♥✜❣✉r❛çã♦ ❞❡❢♦r♠❛❞❛ ❞♦ ♣r♦❜❧❡♠❛ ❝❤❛♣❛ s♦❜ ❝✐s❛❧❤❛♠❡♥t♦ ♣✉r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✼✳✻ ❱✐❣❛ s♦❜ ✢❡①ã♦ ♣✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✼✳✼ ❉✐s❝r❡t✐③❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ s♦❜ ✢❡①ã♦ ♣✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✼✳✽ ❈♦♥✜❣✉r❛çã♦ ❞❡❢♦r♠❛❞❛ ❞♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ s♦❜ ✢❡①ã♦ ♣✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

①✐✈ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

✼✳✾ ❚❡♠♣♦ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✼✳✶✵ ❱✐❣❛ ❡♥❣❛st❛❞❛ s♦❜ ❝❛rr❡❣❛♠❡♥t♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✼✳✶✶ ❉✐s❝r❡t✐③❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ ❡♥❣❛st❛❞❛ s♦❜ ❝❛rr❡❣❛♠❡♥t♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✼✳✶✷ ❈♦♥✜❣✉r❛çã♦ ❞❡❢♦r♠❛❞❛ ❞♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ ❡♥❣❛st❛❞❛ s♦❜ ❝❛rr❡❣❛♠❡♥t♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✻✹ ✼✳✶✸ ❉✐str✐❜✉✐çã♦ ❞❡ t❡♥sõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ ❡♥❣❛st❛❞❛ s♦❜ ❝❛rr❡❣❛♠❡♥t♦ ✉♥✐❢♦r♠❡✳ ✻✺ ✼✳✶✹ ❊rr♦ r❡❧❛t✐✈♦ ❞❛s t❡♥sõ❡s σ12 ♣❛r❛ ✈✐❣❛ ❡♥❣❛st❛❞❛ s♦❜ ❝❛rr❡❣❛♠❡♥t♦ ✉♥✐❢♦r♠❡✳ ✳ ✳ ✳ ✳ ✻✻ ✼✳✶✺ ❋✉r♦ ❡❧í♣t✐❝♦ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✼✳✶✻ ❉✐s❝r❡t✐③❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❢✉r♦ ❡❧í♣t✐❝♦ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✼✳✶✼ ❈♦♥✜❣✉r❛çã♦ ❞❡❢♦r♠❛❞❛ ❞♦ ♣r♦❜❧❡♠❛ ❢✉r♦ ❡❧í♣t✐❝♦ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✼✳✶✽ ❊rr♦ r❡❧❛t✐✈♦ ♣❛r❛ ♦s ❞❡s❧♦❝❛♠❡♥t♦s ❞♦ ♣r♦❜❧❡♠❛ ❢✉r♦ ❡❧í♣t✐❝♦ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✻✽ ✼✳✶✾ ❉✐str✐❜✉✐çã♦ ❞❡ t❡♥sõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❢✉r♦ ❡❧í♣t✐❝♦ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✼✳✷✵ ❉✐s❝r❡t✐③❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❢r❛t✉r❛ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✼✳✷✶ ❈♦♥✜❣✉r❛çã♦ ❞❡❢♦r♠❛❞❛ ❞♦ ♣r♦❜❧❡♠❛ ❢r❛t✉r❛ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✼✳✷✷ ❉✐str✐❜✉✐çã♦ ❞❡ t❡♥sõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❢r❛t✉r❛ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✼✳✷✸ ❚❡♥sõ❡s ❡①❛t❛s ❡ ❛♣r♦①✐♠❛❞❛s ❞❡ σ11 ❡ σ22 ♣❛r❛ ❛ s❡çã♦ ❆✲❆✬ ❞♦ ♣r♦❜❧❡♠❛ ❢r❛t✉r❛

❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✼✳✷✹ ❊rr♦s ❛❜s♦❧✉t♦s ❡ r❡❧❛t✐✈♦s ❞❛s t❡♥sõ❡s σ11 ❡ σ22 ♣❛r❛ ❛ s❡çã♦ ❆✲❆✬ ❞♦ ♣r♦❜❧❡♠❛

❢r❛t✉r❛ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✼✳✷✺ ❚❡♥sõ❡s ❡①❛t❛s ❡ ❛♣r♦①✐♠❛❞❛s ❞❡ σ11 ❡ σ22 ♣❛r❛ ❛ s❡çã♦ ❇✲❇✬ ❞♦ ♣r♦❜❧❡♠❛ ❢r❛t✉r❛

❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✼✳✷✻ ❊rr♦s ❛❜s♦❧✉t♦s ❡ r❡❧❛t✐✈♦s ❞❛s t❡♥sõ❡s σ11 ❡ σ22 ♣❛r❛ ❛ s❡çã♦ ❇✲❇✬ ❞♦ ♣r♦❜❧❡♠❛

❢r❛t✉r❛ ❡♠ ♠❡✐♦ ✐♥✜♥✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

❆✳✶ ❉❡❧t❛ ❞❡ ❉✐r❛❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾

❇✳✶ P❛r❛♠❡tr✐③❛çã♦ ❡❧❡♠❡♥t♦ ❧✐♥❡❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✼✳✶ ❉❡s❧♦❝❛♠❡♥t♦s ♥♦❞❛✐s ❞♦ ♣r♦❜❧❡♠❛ ❝❤❛♣❛ s♦❜ t❡♥sã♦ ✉♥✐❢♦r♠❡ ✭10−05_{♠✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾} ✼✳✷ ❉❡s❧♦❝❛♠❡♥t♦s ♥♦❞❛✐s ❞♦ ♣r♦❜❧❡♠❛ ❝❤❛♣❛ s♦❜ ❝✐s❛❧❤❛♠❡♥t♦ ♣✉r♦ ✭10−05_{♠✮ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵} ✼✳✸ ❉❡s❧♦❝❛♠❡♥t♦s ♥♦❞❛✐s ❞♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ s♦❜ ✢❡①ã♦ ♣✉r❛ ✭10−05_{♠✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷} ✼✳✹ ❚❡♥sõ❡s σ12 ❞♦ ♣r♦❜❧❡♠❛ ✈✐❣❛ ❡♥❣❛st❛❞❛ s♦❜ ❝❛rr❡❣❛♠❡♥t♦ ✉♥✐❢♦r♠❡ ✭▼P❛✮ ✳ ✳ ✳ ✳ ✳ ✻✻

❊✳✶ ❚❡♠♣♦ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ♣❛r❛ ❢♦r♠✉❧❛çã♦ ❡♠ ❞❡s❧♦❝❛♠❡♥t♦ ✭♠s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ❊✳✷ ❚❡♠♣♦ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ♣❛r❛ ❢♦r♠✉❧❛çã♦ ❡♠ ❢♦rç❛ ❞❡ s✉♣❡r❢í❝✐❡ ✭♠s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

➱ ❝r❡s❝❡♥t❡ ❛ ✉t✐❧✐③❛çã♦ ❡str✉t✉r❛❧ ❞❡ ♠❛t❡r✐❛✐s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ❛♥✐s♦tró♣✐❝❛s✳ ■♥✐❝✐❛❧♠❡♥t❡ s❡✉ ❡♠♣r❡❣♦ ♦❝♦rr❡✉ ♥♦ s❡t♦r ❛ér❡♦ ❞❡✈✐❞♦ ❛ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ r✐❣✐❞❡③ ❡ r❡s✐stê♥❝✐❛ r❡❧❛t✐✈❛ ❛ s❡✉ ♣❡s♦✳ ❍♦❥❡✱ ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠❛t❡r✐❛✐s ❜❛r❛t♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ✜❜r❛s ❞❡ ✈✐❞r♦ ❡ r❡s✐♥❛s✱ ♦ ✉s♦ ❞❡st❡s ♠❛t❡r✐❛✐s s❡ ❞✐ss❡♠✐♥♦✉ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛ ✐♥❞ústr✐❛ ❛✉t♦♠♦t✐✈❛✱ ❝♦♥str✉çã♦ ❝✐✈✐❧ ❡ ❛♣❧✐❝❛çã♦ ❡♠ ❛rt✐❣♦s ❡s♣♦rt✐✈♦s ✭❙❝❤❝❧❛r✱ ✶✾✾✹✮✳

P♦rt❛♥t♦✱ ♠ét♦❞♦s ♣r❡❝✐s♦s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❡str✉t✉r❛❧ ❡♠♣r❡❣❛♥❞♦ ❡st❡ t✐♣♦ ❞❡ ♠❛t❡r✐❛❧ é ❝❛❞❛ ✈❡③ ♠❛✐s ♥❡❝❡ssár✐❛✳ ❖ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ❛❞❡q✉❛❞❛ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ❛♥á❧✐s❡✳ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ♣r♦❜❧❡♠❛s ❞❛ ♠❡❝â♥✐❝❛ ❞❛ ✜ss✉r❛✱ ♦ ♠ét♦❞♦ ❛♣r❡s❡♥t❛ ✉♠❛ ✈❛♥t❛❣❡♠ ❡♠ r❡❧❛çã♦ ❛♦s ♦✉tr♦s ♠ét♦❞♦s ♣♦r r❡❞✉③✐r ❛ ❞✐♠❡♥sã♦ ❞♦ ♣r♦❜❧❡♠❛ ❛♥❛❧✐s❛❞♦✳ ❖✉ s❡❥❛✱ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ é ♥❡❝❡ssár✐♦ ❞❡s❝r❡✈❡r s♦♠❡♥t❡ ♦ s❡✉ ❝♦♥t♦r♥♦✱ ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ tr✐❞✐♠❡♥s✐♦♥❛❧ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❞❡s❝r❡✈❡r s♦♠❡♥t❡ s✉❛ s✉♣❡r❢í❝✐❡✳

◆♦ ❡♥t❛♥t♦✱ s✉❛ ✉t✐❧✐③❛çã♦ ❡♥✈♦❧✈❡ ♦ tr❛t❛♠❡♥t♦ ❞❡ ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s ♦✉ q✉❛s❡✲s✐♥❣✉❧❛r❡s✱ ♦ q✉❡ ♠✉✐t❛s ✈❡③❡s é ✈✐st♦ ❝♦♠♦ ✉♠❛ ❞❡s✈❛♥t❛❣❡♠ ❞♦ ♠ét♦❞♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❞❡s❝r❡✈❡✲s❡ ✉♠❛ té❝♥✐❝❛ s✐♠♣❧❡s ❡ ❞✐r❡t❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❞❡ ✐♥t❡❣r❛✐s ✐♠♣ró♣r✐❛s✳ ▼♦str❛✲s❡ t❛♠❜é♠ ❛ té❝♥✐❝❛ ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ✐♥t❡❣r❛çã♦ ♥✉♠ér✐❝❛ ❞❡ ✐♥t❡❣r❛✐s q✉❛s❡✲s✐♥❣✉❧❛r❡s✳

❘❡ss❛❧t❛✲s❡ q✉❡ ♥❡st❡ tr❛❜❛❧❤♦ ❛❞♦t❛✲s❡ ❛ ✜❧♦s♦✜❛ ♣r❡❣❛❞❛ ♣♦r ❇r❡❜❜✐❛ ❡ ❉♦♠✐♥❣✉❡③ ✭✶✾✾✷✮✿ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ tr❛❞✉çã♦ ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ♣❛r❛ ❝♦♥❝❡✐t♦s ❞❡ ❡♥❣❡♥❤❛r✐❛ ❞❡ ❢♦r♠❛ q✉❡ ♦ ♠ét♦❞♦ ♣♦ss❛ s❡r ❝♦♠♣r❡❡♥❞✐❞♦ ❡ ❛♣❧✐❝❛❞♦ ❡♠ s✐t✉❛çõ❡s r❡❛✐s✳

✶✳✶ ❖❜❥❡t✐✈♦s

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ✐♠♣❧❡♠❡♥t❛r ❛ ❢♦r♠✉❧❛çã♦ ❉✉❛❧ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ✉t✐❧✐③❛♥❞♦ ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❛♥✐s♦tró♣✐❝❛✳

✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✸

❆s ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s q✉❡ s✉r❣❡♠ ♥❡st❛ ❢♦r♠✉❧❛çã♦ sã♦ tr❛t❛❞❛s ♣♦r ✉♠❛ té❝♥✐❝❛ ❡s♣❡❝í✜❝❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❞❡ ✐♥t❡❣r❛✐s✱ ♦ ♠ét♦❞♦ ❞❛ s✉❜tr❛çã♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡✳

◆❛ ❞✐s❝r❡t✐③❛çã♦ ❞♦ ❝♦♥t♦r♥♦ sã♦ ❡♠♣r❡❣❛❞♦s ❡❧❡♠❡♥t♦s ✐s♦♣❛r❛♠étr✐❝♦s✱ ♦s q✉❛✐s ♣❡r♠✐t❡♠ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ♦r❞❡♠ ❞❡ ❛♣r♦①✐♠❛çã♦ ❞❛s ❣r❛♥❞❡③❛s ❡♥✈♦❧✈✐❞❛s✳

✶✳✷ ❈♦♥tr✐❜✉✐çõ❡s

❆s ♣r✐♥❝✐♣❛✐s ❝♦♥tr✐❜✉✐çõ❡s ❞❡st❡ tr❛❜❛❧❤♦ sã♦✿

• ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧✳ ❉❡t❛❧❤❛♠❡♥t♦ ❞❛ ❢♦r♠✉❧❛çã♦ ❞✉❛❧ ❡ tr❛t❛♠❡♥t♦ ❞❛s

✐♥t❡❣r❛✐s ✐♠♣ró♣r✐❛s q✉❡ s✉r❣❡♠✳

• ❊❧❡♠❡♥t♦s ✐s♦♣❛r❛♠étr✐❝♦s✳ ❯t✐❧✐③❛çã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ♦r❞❡♠

❞❡ ❛♣r♦①✐♠❛çã♦ ❞❛s ❣r❛♥❞❡③❛s ❡♥✈♦❧✈✐❞❛s✳

• Pr♦❝❡ss❛♠❡♥t♦ ❡♠ ♣❛r❛❧❡❧♦✳ ❖ ❝ó❞✐❣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❡s❡♥✈♦❧✈✐❞♦ ✉t✐❧✐③❛♥❞♦ ♦ ♣❛r❛❞✐❣♠❛ ❞❛

♣r♦❣r❛♠❛çã♦ ♦r✐❡♥t❛❞❛ ❛ ♦❜❥❡t♦s ♣❡r♠✐t❡ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❡♠ ♣❛r❛❧❡❧♦ ❞♦ ♣r♦❜❧❡♠❛ ❛♥❛❧✐s❛❞♦✳

✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦

❆ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ ❞♦s tr❛❜❛❧❤♦s ❝♦♥s✉❧t❛❞♦s ♥❡st❛ ♣❡sq✉✐s❛ é ❛♣r❡s❡♥t❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷✳ ◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛✲s❡ ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❚❡♦r✐❛ ❞❛ ❊❧❛st✐❝✐❞❛❞❡ ▲✐♥❡❛r ❡ ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♠❡✐♦ ❛♥✐s♦tró♣✐❝♦✳

❖ ▼ét♦❞♦ ❞♦s ❘❡sí❞✉♦s P♦♥❞❡r❛❞♦s ❛♣r❡s❡♥t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✹ é ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ❞✐✈❡rs♦s ♠ét♦❞♦s ♥✉♠ér✐❝♦s✳ ❊♠ s❡❣✉✐❞❛✱ é ♠♦str❛❞❛ ❝♦♠♦ ❛s ✐♥t❡❣r❛✐s ❞❡ ❞♦♠í♥✐♦✱ ♦❜t✐❞❛s ♣❡❧♦ ▼ét♦❞♦ ❞♦s ❘❡sí❞✉♦s P♦♥❞❡r❛❞♦s✱ ♣♦❞❡♠ s❡r tr❛♥s❢♦r♠❛❞❛s ❡♠ ✐♥t❡❣r❛✐s ❞❡ ❝♦♥t♦r♥♦✳ ❯♠❛ ✈❡③ q✉❡ t❡♠✲s❡ ♦ ♣r♦❜❧❡♠❛ ❞❡✜♥✐❞♦ ❡♠ ✈❛r✐á✈❡✐s ❞❡ ❝♦♥t♦r♥♦✱ ❛♣r❡s❡♥t❛✲s❡ ♥♦ ❈❛♣ít✉❧♦ ✺ ♦ ♠ét♦❞♦ ♥✉♠ér✐❝♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡st❛s ✐♥t❡❣r❛✐s✱ ♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧✳ ◆❡st❡ ❝❛♣ít✉❧♦ t❛♠❜é♠ ❞✐s❝✉t✐✲s❡ ❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞❛s ✐♥t❡❣r❛✐s q✉❛s❡✲s✐♥❣✉❧❛r❡s ❜❡♠ ❝♦♠♦ ♦ tr❛t❛♠❡♥t♦ ❞❛s ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s ❞❡ ❝♦♥t♦r♥♦✳

◆♦ ❈❛♣ít✉❧♦ ✻✱ ❛♣r❡s❡♥t❛✲s❡ ♦ ❝ó❞✐❣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❡s❡♥✈♦❧✈✐❞♦ ❞❡t❛❧❤❛♥❞♦ s✉❛ ❡str✉t✉r❛ ❡ ✈❛♥t❛❣❡♥s ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ♣r♦❣r❛♠❛çã♦ ♦r✐❡♥t❛❞❛ ❛ ♦❜❥❡t♦s✳ ❆♣r❡s❡♥t❛✲s❡ t❛♠❜é♠ ♦ ❝♦♥❝❡✐t♦ ❞❛ ♣r♦❣r❛♠❛çã♦ ❡♠ ♣❛r❛❧❡❧♦ ❡♠♣r❡❣❛❞❛✳

✶✳✸ ❖❘●❆◆■❩❆➬➹❖ ❉❖ ❚❘❆❇❆▲❍❖ ✸

P♦r ✜♠✱ ❞✐s❝✉t❡✲s❡ ❛❧❣✉♠❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s ♥❡st❡ tr❛❜❛❧❤♦ ❡ s✉❣❡stõ❡s ❞❡ tr❛❜❛❧❤♦s ❢✉t✉r♦s ♥♦ ❈❛♣ít✉❧♦ ✽✳

◆♦ ❆♣ê♥❞✐❝❡ ❆ ❛♣r❡s❡♥t❛✲s❡ ♦ ❝♦♥❝❡✐t♦ ❞♦ ❞❡❧t❛ ❞❡ ❉✐r❛❝ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❡♥❣❡♥❤❛r✐❛ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ♥♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦✳

❖ ♣r♦❝❡ss♦ ❞❡ s✉❜tr❛çã♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ✉♠ ❡❧❡♠❡♥t♦ ❧✐♥❡❛r é ❛♣r❡s❡♥t❛❞♦ ♥♦ ❆♣ê♥❞✐❝❡ ❇ ❡ ❞✐s❝✉t❡✲s❡ ❛❧❣✉♠❛s ❞❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞❛ ❢♦r♠✉❧❛çã♦ ❛♥✐s♦tró♣✐❝❛✳

❈♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ❛❣r❡❣❛r ✐♥❢♦r♠❛çã♦✱ ❞❡t❛❧❤❛✲s❡ ♥♦ ❆♣ê♥❞✐❝❡ ❈ ✉♠❛ ❞❛s té❝♥✐❝❛s ❡①✐st❡♥t❡s ♣❛r❛ s❡ ❛✈❛❧✐❛r ❛s t❡♥sõ❡s s♦❜r❡ ♦ ❝♦♥t♦r♥♦✳

❖s ❝♦♥❝❡✐t♦s ❞❡ ❡rr♦s ❛❜s♦❧✉t♦s ❡ r❡❧❛t✐✈♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ s❡ ❛✈❛❧✐❛r ❛ q✉❛❧✐❞❛❞❡ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❆♣ê♥❞✐❝❡ ❉✳

❈❛♣ít✉❧♦ ✷

❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛

❉❡ ❢♦r♠❛ ♦❜❥❡t✐✈❛✱ ♥❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠✲s❡ ❛s r❡❢❡rê♥❝✐❛s ❝♦♥s✉❧t❛❞❛s✳ ■♥✐❝✐❛❧♠❡♥t❡ sã♦ ❝✐t❛❞♦s ♦s tr❛❜❛❧❤♦s ❞❡ r❡❧❡✈â♥❝✐❛ ♥❛ ❚❡♦r✐❛ ❞❛ ❊❧❛st✐❝✐❞❛❞❡✳ ❉❡♣♦✐s ❛♣r❡s❡♥t❛♠✲s❡ ♦s tr❛❜❛❧❤♦s r❡❧❛❝✐♦♥❛❞♦s à ❢♦r♠✉❧❛çã♦ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧✱ ❜❡♠ ❝♦♠♦ ❛s té❝♥✐❝❛s ✉t✐❧✐✲ ③❛❞❛s ♣❛r❛ s❡ ❛✈❛❧✐❛r ✐♥t❡❣r❛✐s ♥♦ s❡♥t✐❞♦ ♣r✐♥❝✐♣❛❧ ❞❡ ❈❛✉❝❤②✱ ♣❛rt❡ ✜♥✐t❛ ❞❡ ❍❛❞❛♠❛r❞ ❡ ✐♥t❡❣r❛✐s q✉❛s❡✲s✐♥❣✉❧❛r❡s✳ ❈♦♥❝❧✉✐✲s❡ ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠❛ ❜r❡✈❡ ❡①♣♦s✐çã♦ ❞♦s t❡①t♦s q✉❡ ♣♦ss✐❜✐❧✐t❛r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ♣r♦❝❡ss❛♠❡♥t♦ ❡♠ ♣❛r❛❧❡❧♦✳

❆ t❡♦r✐❛ ❞❛ ❡❧❛st✐❝✐❞❛❞❡ ✐s♦tró♣✐❝❛ é t❡♠❛ ❜❛st❛♥t❡ ❡st✉❞❛❞♦ ❡ ♣♦❞❡✲s❡ ❝✐t❛r tr❛❜❛❧❤♦s ❝❧ás✲ s✐❝♦s ♥❡ss❛ ár❡❛ ❝♦♠♦ ▲♦✈❡ ✭✶✾✹✹✮✱ ❚✐♠♦s❤❡♥❦♦ ✭✶✾✼✵✮ ❡ ▼✉s❦❤❡❧✐s❤✈✐❧✐ ✭✶✾✼✼✮✱ ❡ ♠❛✐s r❡❝❡♥t❡ ❈❤♦✉ ❡ P❛❣❛♥♦ ✭✶✾✾✷✮ ❡ ❙❛❞❞ ✭✷✵✵✾✮✳ ◆♦ ❡♥t❛♥t♦✱ ❛ t❡♦r✐❛ ❞❛ ❡❧❛st✐❝✐❞❛❞❡ ❛♥✐s♦tró♣✐❝❛ ♣♦ss✉✐ ♣♦✉✲ ❝❛s✱ ♠❛s ❝♦♠♣❧❡t❛s✱ r❡❢❡rê♥❝✐❛s✳

◆❛ ❧✐t❡r❛t✉r❛ ❡①✐st❡♠ ❞✉❛s ❛❜♦r❞❛❣❡♥s ❞✐❢❡r❡♥t❡s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♣❧❛♥♦s ❞❡ ❡❧❛st✐❝✐❞❛❞❡ ❧✐♥❡❛r ❛♥✐s♦tró♣✐❝❛✿ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ▲❡❦❤♥✐ts❦✐✐ ❡ ♦ ❞❡ ❙tr♦❤✳

▲❡❦❤♥✐ts❦✐✐ ✭✶✾✻✽✱ ✶✾✽✶✮ ❛♣r❡s❡♥t❛ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ▲❡❦❤♥✐ts❦✐✐✱ ♦ q✉❛❧ ✉t✐❧✐③❛ ❛ ❛❜♦r❞❛❣❡♠ ❜❛s❡❛❞❛ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ s♦❧✉çã♦ ❣❡r❛❧ ❛tr❛✈és ❞❡ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ✐♥tr♦❞✉③✐❞❛ ♣♦r ❑♦❧♦s♦✈ ✭✶✾✸✺✮✳ ❆ ❛❜♦r❞❛❣❡♠ ❛❧t❡r♥❛t✐✈❛ à ❞❡ ❑♦❧♦s♦✈ ❡♥✈♦❧✈❡♥❞♦ ❛✉t♦✲✈❛❧♦r❡s✱ ❝♦♥❤❡❝✐❞❛ ♣♦r ❢♦r♠❛❧✐s♠♦ ❞❡ ❙tr♦❤✱ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❙tr♦❤ ✭✶✾✺✽✱ ✶✾✻✷✮ ❡ ❊s❤❡❧❜②✱ ❘❡❛❞✱ ❡ ❙❤♦❝❦❧❡② ✭✶✾✺✸✮ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❞❡t❛❧❤❛❞❛ ❡♠ ❚✐♥❣ ✭✶✾✾✻✮ ❡ ❘❛♥❞ ❡ ❘♦✈❡♥s❦✐ ✭✷✵✵✹✮✳

❍✇✉ ✭✷✵✶✵✮ ❞❡st❛❝❛ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❞✉❛s ❡str❛té❣✐❛s é q✉❡ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ▲❡❦❤♥✐ts❦✐✐ ♣❛rt❡ ❞❛s t❡♥sõ❡s ♣❛r❛ ♦ ❡st❛❞♦ ♣❧❛♥♦✱ ❡♥q✉❛♥t♦ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ❙tr♦❤ ❝♦♠❡ç❛ ❝♦♠ ♦s ❞❡s❧♦❝❛♠❡♥✲ t♦s✳ ❉❡st❛ ❢♦r♠❛✱ ❞❡✈✐❞♦ ❛ ✐♥t❡❣r❛çã♦ ❞❛s ❞❡❢♦r♠❛çõ❡s ♣❧❛♥❛s✱ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ▲❡❦❤♥✐ts❦✐✐ ♣❡r♠✐t❡ r❡❧❛❝✐♦♥❛r ♦s ❞❡s❧♦❝❛♠❡♥t♦s ♣❧❛♥♦s ❝♦♠ ♦ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❝♦❜r✐♥❞♦ ❝❛♠♣♦s ❞❡ ❞❡s✲ ❧♦❝❛♠❡♥t♦s ♠❛✐s ❣❡r❛✐s q✉❡ ♦s ♣❡r♠✐t✐❞♦s ♣❡❧♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ❙tr♦❤✳ ◆♦ ❡♥t❛♥t♦✱ ❍✇✉ r❡ss❛❧t❛ ❛

✻ ❘❊❱■❙➹❖ ❇■❇▲■❖●❘➪❋■❈❆ ✷✳✵

❡❧❡❣â♥❝✐❛ ♠❛t❡♠át✐❝❛ ❞♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ❙tr♦❤ ❝♦♠♦ ✈❛♥t❛❣❡♠ ❞❡st❛ ❡str❛té❣✐❛ ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s ❡♠ q✉❡ ❡❧❛ s❡ ❛♣❧✐❝❛✳

❋✐♥❧❛②s♦♥ ❡ ❙❝r✐✈❡♥ ✭✶✾✻✻✮ ❛♣r❡s❡♥t❛♠ ♦ ♠ét♦❞♦ ❞♦s r❡sí❞✉♦s ♣♦♥❞❡r❛❞♦s ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ s❡ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❛♣r♦①✐♠❛❞❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s q✉❡ ❣♦✈❡r♥❛♠ ✈ár✐♦s t✐♣♦s ❞❡ ♣r♦❜❧❡♠❛s ♥❛ ❡♥❣❡♥❤❛r✐❛✳ ❙❡♥❞♦ t❛♠❜é♠ ❛ té❝♥✐❝❛ ♣❛r❛ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦✳

❈❤❡♥❣ ❡ ❈❤❡♥❣ ✭✷✵✵✺✮ ❛♣r❡s❡♥t❛♠ ♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❡♠ s❡✉ ❝♦♥t❡①t♦ ❤✐stó✲ r✐❝♦ ❞❡s❞❡ ♦s tr❛❜❛❧❤♦s ✐♥✐❝✐❛✐s ❝♦♠ ❡q✉❛çõ❡s ✐♥t❡❣r❛✐s r❡❛❧✐③❛❞♦s ♣♦r ❊r✐❦ ■✈❛r ❋r❡❞❤♦❧♠ ❡♠ ✶✾✵✸✱ ❛té ❛ ❛t✉❛❧✐❞❛❞❡ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ❋r❛♥❦ ❏♦s❡♣❤ ❘✐③③♦✱ ❚❤♦♠❛s ❆❧❧❡♥ ❈r✉s❡ ❡ ❈❛r❧♦s ❆❧❜❡rt♦ ❇r❡❜❜✐❛✳

❙❡❣✉♥❞♦ ❙❝❤❝❧❛r ✭✶✾✾✹✮✱ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ❛♥✐s♦tr♦♣✐❛ ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❞❡ ❞✉❛s ❢♦r♠❛s✿

✶✳ ❙♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ✐s♦tró♣✐❝❛✳ ❊ss❛ ❛❜♦r❞❛❣❡♠ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ✐s♦tró♣✐❝❛ ❞❡ r❡❢❡rê♥❝✐❛ r❡s✉❧t❛♥❞♦ ❡♠ ✐♥t❡❣r❛✐s ❞❡ ❞♦♠í♥✐♦ q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ✉♠ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ ♣❛r❛ s❡ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❝♦rr❡t♦s ❞❡ ❞❡s❧♦❝❛♠❡♥t♦s ❡ ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡✳ ❆❧✲ t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ❛♣❧✐❝❛r ♦ ▼ét♦❞♦ ❞❛ ❘❡❝✐♣r♦❝✐❞❛❞❡ ❉✉❛❧ ✭P❛rtr✐❞❣❡ ❡t ❛❧✳✱ ✶✾✾✷✮ às ✐♥t❡❣r❛✐s ❞❡ ❞♦♠í♥✐♦ ♣❛r❛ s❡ tr❛❜❛❧❤❛r s♦♠❡♥t❡ ❝♦♠ ✐♥t❡❣r❛✐s ❞❡ ❝♦♥t♦r♥♦❀

✷✳ ❙♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❛♥✐s♦tró♣✐❝❛✳ ❊ss❛ ❛❜♦r❞❛❣❡♠ é ❞❡ ❛♣❧✐❝❛çã♦ ♥❛t✉r❛❧ ❡ ✐♠❡❞✐❛t❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡♥✈♦❧✈❡ ♦ ❡♠♣r❡❣♦ ❞❡ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s✳

❯♠ ❞♦s ♣r✐♠❡✐r♦s tr❛❜❛❧❤♦s ❛ tr❛t❛r ❞❡ ♣r♦❜❧❡♠❛s ♥ã♦ ✐s♦tró♣✐❝♦s ❢♦✐ ♦ ❞❡ ❘✐③③♦ ❡ ❙❤✐♣♣② ✭✶✾✼✵✮✳ ❉❡♣♦✐s ❞❡❧❡✱ ♦ tr❛❜❛❧❤♦ ❞❡ ❈r✉s❡ ❡ ❙✇❡❞❧♦✇ ✭✶✾✼✶✮ ❡♠♣r❡❣❛♥❞♦ ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❛♥✐s♦tró♣✐❝❛ ❞❡✉ ♦r✐❣❡♠ ❛ ✉♠❛ sér✐❡ ❞❡ tr❛❜❛❧❤♦s s❡❣✉✐♥❞♦ ❛ ♠❡s♠❛ ❧✐♥❤❛ ❝♦♠♦ ❙♥②❞❡r ❡ ❈r✉s❡ ✭✶✾✼✺✮✱ ❈❤♦ ❡t ❛❧✳ ✭✶✾✾✷✮✱ ❚❛♥ ❡ ●❛♦ ✭✶✾✾✷✮✱ ❉❡♥❞❛ ❡ ▼❛r❛♥t❡ ✭✷✵✵✹✮ ❡ ◆♦✉r✐♥❡ ❡t ❛❧✳ ✭✷✵✶✵✮✳

❙♦❧❧❡r♦ ❡ ❆❧✐❛❜❛❞✐ ✭✶✾✾✸✮ ❥á ❛♣r❡s❡♥t❛✈❛♠ t♦❞♦s ♦s ❢✉♥❞❛♠❡♥t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❢♦r♠✉❧❛✲ çã♦ ❤✐♣❡r✲s✐♥❣✉❧❛r ❞♦ ♣r♦❜❧❡♠❛ ♣❧❛♥♦ ❛♥✐s♦tró♣✐❝♦✳ ▼❛s s♦♠❡♥t❡ ❡♠ ❙♦❧❧❡r♦ ❡ ❆❧✐❛❜❛❞✐ ✭✶✾✾✺✮ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❢♦r♠❛❧♠❡♥t❡ ♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧ ❆♥✐s♦tró♣✐❝♦✳

✷✳✵ ✼

◆❡st❡ tr❛❜❛❧❤♦ ❛❞♦t❛✲s❡ ❛ té❝♥✐❝❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❞❡ ✐♥t❡❣r❛✐s ♣❡❧♦ ♠ét♦❞♦ ❞❡ s✉❜tr❛çã♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡ ✭❙❧á❞❡❦ ❡ ❙❧❛❞❡❦✱ ✶✾✾✽✮ ♣❛r❛ tr❛t❛r ❞❛s ✐♥t❡❣r❛✐s s✐♥❣✉❧❛r❡s✳ P❛r❛ ❛s q✉❛s❡✲s✐♥❣✉❧❛r❡s✱ ❛ té❝♥✐❝❛ ❛❞♦t❛❞❛ ❢♦✐ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛✉t♦✲❛❞❛♣t❛t✐✈❛ ♣r♦♣♦st❛ ♣♦r ❚❡❧❧❡s ✭✶✾✽✼✮✳ ❊♠ q✉❛❧q✉❡r ♠ét♦❞♦ ♥✉♠ér✐❝♦ q✉❡ s❡ ❞❡s❡♥✈♦❧✈❛ é ♥❡❝❡ssár✐♦ t❡r ♣❛râ♠❡tr♦s ♣❛r❛ ❛✈❛❧✐❛r s✉❛ ♣r❡❝✐sã♦✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ❢❛③❡r ✐ss♦ é ❝♦♠♣❛r❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s ❝♦♠ ❛s s♦❧✉çõ❡s ❛♥❛❧í✲ t✐❝❛s ❞✐s♣♦♥í✈❡✐s✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❡♥sõ❡s✱ ❡st❛s s♦❧✉çõ❡s t♦r♥❛♠✲s❡ ❡s❝❛ss❛s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❙❛✈✐♥ ✭✶✾✻✶✮ ❛♣r❡s❡♥t❛ ✉♠ ❡①t❡♥s♦ tr❛❜❛❧❤♦ ❝♦♠ s♦❧✉çõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ t❡♥sõ❡s ❛♦ r❡❞♦r ❞❡ ❢✉r♦s ♣❛r❛ ♠❡✐♦s ❛♥✐s♦tró♣✐❝♦s✳

❈❛♣ít✉❧♦ ✸

❚❡♦r✐❛ ❞❛ ❊❧❛st✐❝✐❞❛❞❡

◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ ❜r❡✈❡ ❡①♣♦s✐çã♦ ❞❛ ❚❡♦r✐❛ ❞❛ ❊❧❛st✐❝✐❞❛❞❡ ▲✐♥❡❛r s❡❣✉✐♥❞♦ ❛ ♠❡t♦❞♦❧♦❣✐❛ ✉t✐❧✐③❛❞❛ ♣♦r ❚✐♠♦s❤❡♥❦♦ ✭✶✾✼✵✮ ❡ ❚✐♥❣ ✭✶✾✾✻✮✳ ❊ss❛ ✐♥tr♦❞✉çã♦ ♣❡r♠✐t✐rá ♦ ❡♥t❡♥✲ ❞✐♠❡♥t♦ ❡ ❝♦♥str✉çã♦ ❞♦ ▼ét♦❞♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❈♦♥t♦r♥♦ ❉✉❛❧ ❆♥✐s♦tró♣✐❝♦✱ q✉❡ é ♦ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦✳

✸✳✶ ❉❡❢♦r♠❛çã♦

❈♦♥s✐❞❡r❡ q✉❡ ♦ ♣♦♥t♦ P ❝♦♠ ❝♦♦r❞❡♥❛❞❛s (X1, X2) ♥♦ ❡st❛❞♦ ✐♥❞❡❢♦r♠❛❞♦ s❡❥❛ ❞❡s❧♦❝❛❞♦ ♣❛r❛ ♦ ♣♦♥t♦ P′ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s

(x1, x2) ♣♦r ✉♠❛ ❞❡❢♦r♠❛çã♦ ❞♦ ❝♦r♣♦ ✭❋✐❣✉r❛ ✸✳✶✮✳ ❖ ✈❡t♦r ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ♣♦♥t♦P t❡♠ ❝♦♠♦ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s(u1, u2) ❡♠ q✉❡

u1=x1−X1 ✭✸✳✶❛✮

u2=x2−X2 ✭✸✳✶❜✮

❈♦♥s✐❞❡r❡ ✉♠ ❡❧❡♠❡♥t♦ r❡t❛♥❣✉❧❛r P QRS ♥♦ ❡st❛❞♦ ✐♥❞❡❢♦r♠❛❞♦✱ ❝♦♠ ❧❛❞♦s(dX1, dX2) ♣❛r❛✲ ❧❡❧♦ ❛♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✭❋✐❣✉r❛ ✸✳✷✮✳ ❉❡♣♦✐s ❞❡ ❞❡❢♦r♠❛❞♦✱ ♦s ♣♦♥t♦s P✱ Q✱ R ❡ S ♠♦✈❡♠✲s❡

r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣❛r❛P′✱ Q′✱

R′❡

S′✳ ❆s ❝♦♦r❞❡♥❛❞❛s ❞❡

Qr❡❧❛t✐✈♦ ❛Psã♦(dX1,0)❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ Q′ r❡❧❛t✐✈♦ ❛

P′ sã♦

(dx1, dx2)✱ ❡♠ q✉❡

dx1 =dX1+du1 =dX1+

∂u1

∂X1

dX1 ✭✸✳✷❛✮

dx2 =du2 =

∂u2

∂X1

dX1 ✭✸✳✷❜✮

✶✵ ❚❊❖❘■❆ ❉❆ ❊▲❆❙❚■❈■❉❆❉❊ ✸✳✶

u2

u1

X1

x1 X2

x2 P

P’

x2

x1

❋✐❣✉r❛ ✸✳✶✿ ❱❡t♦r ❞❡s❧♦❝❛♠❡♥t♦✳

dX

2

dX1

x2

x1 S

P Q

R

P’

Q’

S’ R’

g1

g2

dX1+ dX_∂X∂u1 1 1

dX1

∂u2

∂X1

❋✐❣✉r❛ ✸✳✷✿ ❉❡❢♦r♠❛çã♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧✳

❚❡♠✲s❡✱ ♣♦rt❛♥t♦✱

P′ Q′

=

q dx2

1+dx22

≈dX1+

∂u1

∂X1

dX1 ✭✸✳✸✮

❊♠ q✉❡ ❢♦r❛♠ ❞❡s♣r❡③❛❞♦s ♦s ❡❧❡♠❡♥t♦s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♦r❞❡♠ ✐❣✉❛❧ ❡ s✉♣❡r✐♦r ❛ ✷✱ ❝♦♥❢♦r♠❡ ❤✐♣ót❡s❡ ❞❛ t❡♦r✐❛ ❞❡ ♣❡q✉❡♥♦s ❞❡s❧♦❝❛♠❡♥t♦s ❡ ♣❡q✉❡♥❛s ❞❡❢♦r♠❛çõ❡s✳ P♦rt❛♥t♦✱ ♦ ✐♥❝r❡♠❡♥t♦ ♥♦ ❝♦♠♣r✐♠❡♥t♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❡❧❡♠❡♥t♦ P Q✱ ❞❡♥♦t❛❞♦ ♣♦r ε11✱ é ❞❛❞♦ ♣♦r

ε11=

P′ Q′

−P Q

P Q =

∂u1

✸✳✶ ❉❊❋❖❘▼❆➬➹❖ ✶✶

❉❡ ❢♦r♠❛ s✐♠✐❧❛r✱ ♦ ✐♥❝r❡♠❡♥t♦ ♥♦ ❝♦♠♣r✐♠❡♥t♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❡❧❡♠❡♥t♦ P S

é

ε22=

∂u2

∂X2 ✭✸✳✺✮

❆s ❣r❛♥❞❡③❛s ε11 ❡ ε22 sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❞❡❢♦r♠❛çõ❡s ♥♦r♠❛✐s✳ ❙❡❥❛ γ1 ♦ â♥❣✉❧♦ ❡♥tr❡P′Q′ ❡ ♦ ❡✐①♦ ❝♦♦r❞❡♥❛❞♦x1✱ t❡♠✲s❡

tanγ1=

∂u2/∂X1

1 +∂u1/∂X1 ✭✸✳✻✮

❈♦♥s✐❞❡r❛♥❞♦✲s❡ q✉❡ ♦ â♥❣✉❧♦γ1✱ ♠❡❞✐❞♦ ❡♠ r❛❞✐❛♥♦s✱ s❡❥❛ ♣❡q✉❡♥♦ ❡ ❞❡s♣r❡③❛♥❞♦ ♦s ❡❧❡♠❡♥t♦s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡ s✉♣❡r✐♦r❡s✱ t❡♠✲s❡

γ1=

∂u2

∂X1 ✭✸✳✼✮

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♦ â♥❣✉❧♦ γ2 s❡rá

γ2=

∂u1

∂X2 ✭✸✳✽✮

❙❡❥❛ 2ε12 ♦ ❞❡❝rés❝✐♠♦ ♥♦ â♥❣✉❧♦ ❡♥tr❡P Q❡ P S✱ t❡♠✲s❡

2ε12=γ1+γ2 =

∂u2

∂X1

+ ∂u1

∂X2 ✭✸✳✾✮

❆ ❣r❛♥❞❡③❛ ε12 é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡❢♦r♠❛çã♦ ❛♥❣✉❧❛r✳ ❊ ♦ sí♠❜♦❧♦ γ12 = 2ε12 é ❝♦♠✉♠❡♥t❡ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡♥♦t❛r ❛ ❞❡❢♦r♠❛çã♦ t❛♥❣❡♥t❡ ❡♠ ❛♣❧✐❝❛çõ❡s ❞❡ ❡♥❣❡♥❤❛r✐❛✳

P♦rt❛♥t♦✱ t❡♠✲s❡ ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧ tr✐❞✐♠❡♥s✐♦♥❛❧ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ❞❡❢♦r♠❛çã♦✲❞❡s❧♦❝❛♠❡♥t♦

ε11=

∂u1

∂X1 ✭✸✳✶✵❛✮

ε22=

∂u2

∂X2 ✭✸✳✶✵❜✮

ε33=

∂u3

∂X3 ✭✸✳✶✵❝✮

ε12=

γ12

2 = 1 2

∂u2

∂X1

+ ∂u1

∂X2

✭✸✳✶✵❞✮

ε13=

γ13

2 = 1 2

∂u3

∂X1

+ ∂u1

∂X3

✭✸✳✶✵❡✮

ε23=

γ23

2 = 1 2

∂u3

∂X2

+ ∂u2

∂X3

✶✷ ❚❊❖❘■❆ ❉❆ ❊▲❆❙❚■❈■❉❆❉❊ ✸✳✷

❆s q✉❛✐s ♣♦❞❡♠ s❡r r❡❡s❝r✐t❛s ✉t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦ ✐♥❞✐❝✐❛❧

εij =

1

2(ui,j+uj,i) i, j= 1,2,3 ✭✸✳✶✶✮

✸✳✷ ❚❡♥sã♦

❯♠ ❝♦r♣♦ ♣♦❞❡ s❡r s♦❧✐❝✐t❛❞♦ ♣♦r ❞♦✐s t✐♣♦s ❞❡ ❢♦rç❛s ❡①t❡r♥❛s✿ ❢♦rç❛s ❞❡ ❝♦r♣♦ ❡ ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡✳ ❆s ❢♦rç❛s ❞❡ ❝♦r♣♦ ❛t✉❛♠ s♦❜r❡ t♦❞♦ ♦ ✈♦❧✉♠❡ ❞♦ ❝♦r♣♦✱ ❡♥q✉❛♥t♦ ❛s ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡ ❛t✉❛♠ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ♠❡s♠♦✳ ❊①✐st❡♠ t❛♠❜é♠ ❛s ❢♦rç❛s ✐♥t❡r♥❛s✱ q✉❡ ❛t✉❛♠ ♥❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛s ♣❛rt❡s ❞♦ ❝♦r♣♦ q✉❛♥❞♦ ❡st❡ é s✉❜♠❡t✐❞♦ à ❞❡❢♦r♠❛çã♦✳

❈♦♥s✐❞❡r❡ ♣❛rt❡ ❞❡ ✉♠ só❧✐❞♦ ♦❝✉♣❛♥❞♦ ✉♠❛ r❡❣✐ã♦ V ❞❡❧✐♠✐t❛❞❛ ♣❡❧❛ s✉♣❡r❢í❝✐❡ S ♥♦ ❡st❛❞♦

❞❡❢♦r♠❛❞♦ ✭❋✐❣✉r❛ ✸✳✸✮✳ ❙♦❜r❡ ♦ ❝♦♥t♦r♥♦ S ❛t✉❛♠ ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡ ❝❛✉s❛❞❛s ♣❡❧❛ ❛çã♦ ❞♦

♠❛t❡r✐❛❧ ❡①t❡r✐♦r ❛V s♦❜r❡ ♦ ♠❛t❡r✐❛❧ ✐♥t❡r✐♦r ❛V✳ ❆ t❡♥sã♦ ❛t✉❛♥t❡ ♥❛ ár❡❛δS ♥♦ ♣♦♥t♦P s♦❜r❡

S é ❞❛❞❛ ♣♦r

lim

δS→0

δ~p δS

✭✸✳✶✷✮

❊♠ q✉❡ δ~p é ♦ ✈❡t♦r ❞❡ ❢♦rç❛ q✉❡ ❛t✉❛ s♦❜r❡ ♦ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ár❡❛ δS✳ ❊ss❡ ❧✐♠✐t❡ é

❝♦♥❤❡❝✐❞♦ ❝♦♠♦ tr❛❝t✐♦♥ ♦✉ ✈❡t♦r t❡♥sã♦✱ ❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s ♣♦❞❡♠ s❡r ❞❡❝♦♠♣♦st❛s ❡♠ ♥♦r♠❛✐s ❡ ❝✐s❛❧❤❛♥t❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠ ✈❡t♦r ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡ ❡♠ q✉❡stã♦✳

R

V

S

P

d

S

n

d

p

x2

x1 x3

❋✐❣✉r❛ ✸✳✸✿ ❋♦rç❛ ❞❡ s✉♣❡r❢í❝✐❡✳

❈♦♥s✐❞❡r❡ ✉♠ ❡❧❡♠❡♥t♦ ❝ú❜✐❝♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✭❋✐❣✉r❛ ✸✳✹✮✳ ❆s ❝♦♠✲ ♣♦♥❡♥t❡s ❞♦ ✈❡t♦r t❡♥sã♦ ❛t✉❛♥❞♦ ♥❛ ❢❛❝❡ ❝♦♠ ✈❡t♦r ♥♦r♠❛❧ ❛ ❞✐r❡çã♦ ♣♦s✐t✐✈❛ ❞♦ ❡✐①♦x1 sã♦ σ11✱

σ12 ❡σ13✳ P♦rt❛♥t♦ ♦ ♣r✐♠❡✐r♦ í♥❞✐❝❡ ✐♥❞✐❝❛ ❛ ❞✐r❡çã♦ ❞♦ ✈❡t♦r ♥♦r♠❛❧ ❞❛ ❢❛❝❡ ❞♦ ❝✉❜♦ ❡♠ q✉❡ ❛t✉❛

✸✳✸ ❊◗❯❆➬Õ❊❙ ❉❊ ❊◗❯■▲❮❇❘■❖ ✶✸ s12 s11 s13 s32 s31 s33 s22 s21 s23 x1 x3 x2

❋✐❣✉r❛ ✸✳✹✿ ❈♦♠♣♦♥❡♥t❡s ❞❡ t❡♥sã♦✳

P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ t❡♥sã♦ ❡♠ s✉❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✿

σ=      

σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

      ✭✸✳✶✸✮

✸✳✸ ❊q✉❛çõ❡s ❞❡ ❊q✉✐❧í❜r✐♦

❈♦♥s✐❞❡r❡ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ✉♠ ❝✉❜♦ ✐♥✜♥✐t❡s✐♠❛❧✱ ❝❡♥tr♦ ❡♠ P(x1, x2, x3) ❡ ❞❡ ❧❛❞♦s δx1✱δx2 ❡

δx3 ♣❛r❛❧❡❧♦s ❛♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ✭❋✐❣✉r❛ ✸✳✺✮✳ ❖s ❝❡♥tr♦s ❞❛s ✻ ❢❛❝❡s ❞♦ ❝✉❜♦ sã♦ ♦s ♣♦♥t♦s

(x1±δx1, x2, x3), (x1, x2±δx2, x3), (x1, x2, x3±δx3) ✭✸✳✶✹✮

❙❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ t❡♥sã♦ ♥♦ ♣♦♥t♦ P sã♦ σ11✱σ12✱σ13✱σ21✱σ22✱σ23✱ σ31✱ σ32 ❡ σ33✱ ❡♥tã♦ ❛s ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡ q✉❡ ❛t✉❛♠ ♥❛ ❢❛❝❡ ❝♦♠ ❝❡♥tr♦ ❡♠(x1+δx1, x2, x3)sã♦

σ11+

1 2

∂σ11

∂x1

δx1, σ12+

1 2

∂σ12

∂x1

δx1, σ13+

1 2

∂σ13

∂x1

δx1 ✭✸✳✶✺✮

❊ ❛s ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡ q✉❡ ❛t✉❛♠ ♥❛ ❢❛❝❡ ❝♦♠ ❝❡♥tr♦ ❡♠ (x1−δx1, x2, x3)sã♦

−

σ11−

1 2 ∂σ11 ∂x1 δx1 , ₋ σ12−

1 2 ∂σ12 ∂x1 δx1 , ₋ σ13−

✶✹ ❚❊❖❘■❆ ❉❆ ❊▲❆❙❚■❈■❉❆❉❊ ✸✳✸

x1

x3 x2

dx1

dx3 dx2

(x1 - , x2, x3) 2 dx1

(s11-1ds11)dx2dx3

2 (s11+ ds11)dx2dx3

1 2 (x1 + , x2, x3)

2 dx1

❋✐❣✉r❛ ✸✳✺✿ ❊q✉✐❧í❜r✐♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡✳

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ❛s ❢♦rç❛s ❞❡ s✉♣❡r❢í❝✐❡ ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ♣❛r❛ ❛s ♦✉tr❛s ❢❛❝❡s✳ ❙❡(f1, f2, f3) ❞❡♥♦t❛♠ ❛s ❢♦rç❛s ❡①t❡r♥❛s ❞❡ ❝♦r♣♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡✱ ❡♥tã♦ s♦♠❛♥❞♦ ❛s ❢♦rç❛s q✉❡ ❛t✉❛♠ ♥❛ ❞✐r❡çã♦ ❞❡x1✱ t❡♠✲s❡

σ11+

1 2 ∂σ11 ∂x1 δx1

δx2δx3−

σ11−

1 2 ∂σ11 ∂x1 δx1 δx2δx3

+

σ21+

1 2 ∂σ21 ∂x2 δx2

δx1δx3−

σ21−

1 2 ∂σ21 ∂x2 δx2 δx1δx3

+

σ31+

1 2 ∂σ31 ∂x3 δx3

δx1δx2−

σ31−

1 2 ∂σ31 ∂x3 δx3 δx1δx2

+f1δx1δx2δx3 = 0

✭✸✳✶✼✮

❉✐✈✐❞✐♥❞♦ ♣♦r δx1✱δx2 ❡ δx3✱ t❡♠✲s❡

∂σ11

∂x1

+∂σ21

∂x2

+∂σ31

∂x3

+f1 = 0 ✭✸✳✶✽✮

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣❛r❛ ♦s ❡✐①♦s x2 ❡x3 t❡♠✲s❡

∂σ12

∂x1

+∂σ22

∂x2

+∂σ32

∂x3

+f2 = 0 ✭✸✳✶✾✮

∂σ13

∂x1

+∂σ23

∂x2

+∂σ33

∂x3

+f3 = 0 ✭✸✳✷✵✮

❊st❛s ❡q✉❛çõ❡s sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❡q✉❛çõ❡s ❞❡ ❡q✉✐❧í❜r✐♦ ❡♠ t❡♥sã♦ ❞❡ ✉♠ só❧✐❞♦ ❡ ♣♦❞❡♠ s❡r r❡❡s❝r✐t❛s ❡♠ ♥♦t❛çã♦ ✐♥❞✐❝✐❛❧

σij,j +fi = 0 i, j= 1,2,3 ✭✸✳✷✶✮

✸✳✸ ❊◗❯❆➬Õ❊❙ ❉❊ ❊◗❯■▲❮❇❘■❖ ✶✺

❞❡ ♠♦♠❡♥t♦s✳ ❋❛③❡♥❞♦ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ♠♦♠❡♥t♦s ♥♦ ❡✐①♦ ♣❛r❛❧❡❧♦ ❛♦ x1 ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ P ❡♥❝♦♥tr❛✲s❡

1 2δx2

σ23+

1 2

∂σ23

∂x2

δx2

δx1δx3+

1 2δx2

σ23−

1 2

∂σ23

∂x2

δx2

δx1δx3

− 1₂δx3

σ32+

1 2

∂σ32

∂x3

δx3

δx1δx2+

1 2δx3

σ32−

1 2

∂σ32

∂x3

δx3

δx1δx2 = 0

✭✸✳✷✷✮

❉✐✈✐❞✐♥❞♦ ♣♦r δx1✱δx2 ❡ δx3✱ t❡♠✲s❡

σ23=σ32 ✭✸✳✷✸✮

❉❡ ❢♦r♠❛ s✐♠✐❧❛r✱ ❢❛③❡♥❞♦ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ♠♦♠❡♥t♦s ❡♠ r❡❧❛çã♦ ❛♦s ❡✐①♦s ♣❛r❛❧❡❧♦s ❛ x2 ❡ x3 ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ P é ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡

σ12=σ21 ✭✸✳✷✹✮

σ13=σ31 ✭✸✳✷✺✮

❊ss❛s r❡❧❛çõ❡s ❞❡ s✐♠❡tr✐❛ ✐♠♣❧✐❝❛♠ q✉❡ ❞❛s ✾ ❝♦♠♣♦♥❡♥t❡s ❞❡ t❡♥sã♦ ❡①✐st❡♥t❡s✱ s♦♠❡♥t❡ ✻ ❞❡❧❛s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳

➱ ♣♦ssí✈❡❧ t❛♠❜é♠ ♣r♦✈❛r q✉❡ ❛ ❢♦rç❛ ❞❡ s✉♣❡r❢í❝✐❡ ❡♠ ✉♠ ♣♦♥t♦ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❛r❜✐trár✐❛ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ t❡♥sã♦ ♥❛q✉❡❧❡ ♣♦♥t♦✳ P❛r❛ ❡❢❡✐t♦ ❞❡ ❞❡♠♦♥str❛çã♦✱ ❝♦♥s✐❞❡r❡ q✉❡

σ31=σ32=σ33= 0 ✭✸✳✷✻✮

❈♦♥s✐❞❡r❡ ♦ ❡❧❡♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧ ❞❛ ❋✐❣✉r❛ ✭✸✳✻✮✳ ❋❛③❡♥❞♦ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ❢♦rç❛s ♥♦ ❡✐①♦ x1✱ t❡♠✲s❡

t1AB−σ11OB−σ21OA= 0 ✭✸✳✷✼✮

◆♦t❡ q✉❡ OA=ABsinθ ❡OB =ABcosθ✱ ❡♥tã♦

✶✻ ❚❊❖❘■❆ ❉❆ ❊▲❆❙❚■❈■❉❆❉❊ ✸✳✹ t1 s11 s12 s12 s11 O B A

t2 n

q

x2

x1

❋✐❣✉r❛ ✸✳✻✿ ❊q✉✐❧í❜r✐♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ♣r✐s♠át✐❝♦✳

❉❡ ❢♦r♠❛ s✐♠✐❧❛r✱ t❡♠✲s❡

t2 =σ12cosθ+σ22sinθ ✭✸✳✷✾✮

t3 = 0 ✭✸✳✸✵✮

❊st❛s ❡q✉❛çõ❡s ♣♦❞❡♠ s❡r ❡①♣r❡ss❛s ❡♠ ❢♦r♠❛ ♠❛tr✐❝✐❛❧

           t1 t2 t3            =      

σ11 σ12 0

σ21 σ22 0

0 0 1

                 n1 n2 n3            ✭✸✳✸✶✮

❖✉ ❡♠ ♥♦t❛çã♦ ✐♥❞✐❝✐❛❧

ti =σijnj i, j= 1,2,3 ✭✸✳✸✷✮

❊♠ q✉❡ (n1, n2, n3) = (cosθ,sinθ,0)sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡✳

✸✳✹ ❊q✉❛çõ❡s ❞❡ ❈♦♠♣❛t✐❜✐❧✐❞❛❞❡

❈♦♥s✐❞❡r❡ ❛ r❡❧❛çã♦ ❞❡❢♦r♠❛çã♦✲❞❡s❧♦❝❛♠❡♥t♦ ✈á❧✐❞❛ ♣❛r❛ ♣❡q✉❡♥❛s ❞❡❢♦r♠❛çõ❡s

εij =

1

2(ui,j+uj,i) i, j= 1,2,3