Condições essenciais de contorno e interface na análise de cascas com métodos sem...

❏❖❘●❊ ❈❆❘❱❆▲❍❖ ❈❖❙❚❆

❊❙❙❊◆❚■❆▲ ❇❖❯◆❉❆❘❨ ❆◆❉ ■◆❚❊❘❋❆❈❊ ❈❖◆❉■❚■❖◆❙ ■◆ ❚❍❊ ▼❊❙❍▲❊❙❙ ❆◆❆▲❨❙■❙ ❖❋ ❙❍❊▲▲❙

❏❖❘●❊ ❈❆❘❱❆▲❍❖ ❈❖❙❚❆

❊❙❙❊◆❚■❆▲ ❇❖❯◆❉❆❘❨ ❆◆❉ ■◆❚❊❘❋❆❈❊ ❈❖◆❉■❚■❖◆❙ ■◆ ❚❍❊ ▼❊❙❍▲❊❙❙ ❆◆❆▲❨❙■❙ ❖❋ ❙❍❊▲▲❙

❚❤❡s✐s s✉❜♠✐tt❡❞ t♦ t❤❡ P♦❧②t❡❝❤♥✐❝ ❙❝❤♦♦❧ ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦ ❢♦r ❛✇❛r❞ ♦❢ t❤❡ ❉♦❝t♦r ♦❢ ❙❝✐❡♥❝❡ ❉❡❣r❡❡✳

❏❖❘●❊ ❈❆❘❱❆▲❍❖ ❈❖❙❚❆

❊❙❙❊◆❚■❆▲ ❇❖❯◆❉❆❘❨ ❆◆❉ ■◆❚❊❘❋❆❈❊ ❈❖◆❉■❚■❖◆❙ ■◆ ❚❍❊ ▼❊❙❍▲❊❙❙ ❆◆❆▲❨❙■❙ ❖❋ ❙❍❊▲▲❙

❚❤❡s✐s s✉❜♠✐tt❡❞ t♦ t❤❡ P♦❧②t❡❝❤♥✐❝ ❙❝❤♦♦❧ ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦ ❢♦r ❛✇❛r❞ t❤❡ ❉♦❝t♦r ♦❢ ❙❝✐❡♥❝❡ ❉❡❣r❡❡✳

❆r❡❛✿ ❙tr✉❝t✉r❛❧ ❊♥❣✐♥❡❡r✐♥❣

❆❞✈✐s♦r✿ Pr♦❢✳ ❉r✳P❛✉❧♦ ❞❡ ▼❛tt♦s P✐♠❡♥t❛

Catalogação-na-publicação

Costa, Jorge

Essential boundary and interface conditions in the meshless analysis of shells / J. Costa -- São Paulo, 2015.

63 p.

Tese (Doutorado) - Escola Politécnica da Universidade de São Paulo. Departamento de Engenharia de Estruturas e Geotécnica.

❊ss❡♥t✐❛❧ ❜♦✉♥❞❛r② ❛♥❞ ✐♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡

♠❡s❤❧❡ss ❛♥❛❧②s✐s ♦❢ s❤❡❧❧s

❆❝❦♥♦✇❧❡❞❣♠❡♥ts

❋✐rst❧②✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ Pr♦❢✳ P❛✉❧♦ P✐♠❡♥t❛ ❢♦r t❛❦✐♥❣ ♠❡ ❛s ❛ st✉❞❡♥t ❛♥❞ ❣✉✐❞✐♥❣ ♠❡ t❤r♦✉❣❤ ♠② r❡s❡❛r❝❤ ❛♥❞ ♠♦r❡ ✐♠♣♦rt❛♥t❧② t❤r♦✉❣❤ ❛ r❡❛❧ ❛❝❛❞❡♠✐❝ ❢♦r♠❛t✐♦♥✳ ❚❤❛♥❦ ②♦✉ ❢♦r t❤❡ ♠❛♥② ❢r✉✐t❢✉❧❧ ❛♥❞ ❤❛♣♣② ❣❡t✲t♦❣❡t❤❡rs ❛♥❞ ❝♦♥✈❡rs❛t✐♦♥s✱ ❛❜♦✉t ♠② r❡s❡❛r❝❤ ❛♥❞ ❛ ✇❤♦❧❡ ✈❛r✐❛t② ♦❢ ♦t❤❡r s✉❜❥❡❝ts✳ ■t ❤❛s ❜❡❡♥ ❛♥ ❤♦♥♦r t♦ ❜❡ ②♦✉r st✉❞❡♥t✳

❖t❤❡r ♠❛st❡rs ✇❡r❡ ❛❧s♦ ❦✐♥❞ ❡♥♦✉❣❤ t♦ ❛❞✈✐s❡ ❛♥❞ ❤❡❧♣ ♠❡✱ ❡s♣❡❝✐❛❧❧② Pr♦❢✳ ❈❛r❧♦s ❚✐❛❣♦ ✇❤♦ ✇❡❧❝♦♠❡❞ ♠❡ ❢♦r ❛ st❛② ✐♥ ▲✐s❜♦♥ ❞✉r✐♥❣ ♠② ♠❛st❡rs ❛♥❞ ❜❡❝❛♠❡ ♠② r❡❢❡rr❡♥❝❡ ❢♦r ❛❧❧ ♠❛tt❡rs ♦❢ ♠❡s❤❧❡ss ♠❡t❤♦❞s✱ ❛♥❞ Pr♦❢✳ ❊❞✉❛r❞♦ ❈❛♠♣❡❧❧♦ ❢♦r ❣r❡❛t ✐♥s✐❣❤ts ❛♥❞ ❤❡❧♣❢✉❧ ❞✐s❝✉ss✐♦♥s✳

▼② s✐♥❝❡r❡ ❣r❛t✐t✉t❡ t♦ Pr♦❢✳ P❡t❡r ❲r✐❣❣❡rs ❢♦r r❡❝❡✐✈✐♥❣ ♠❡ ❛t t❤❡ ■♥st✐t✉t ❢ür ❑♦♥✲ t✐♥✉✉♠s♠❡❝❤❛♥✐❦ ✐♥ ❍❛♥♥♦✈❡r ❢♦r ❛ ♣r♦❧✐✜❝ ❛♥❞ ✈❡r② ♣❧❡❛s❛♥t ②❡❛r✳ ❚❤❛♥❦ ②♦✉ ❢♦r ❤❡❧♣✐♥❣ ♠② r❡s❡❛r❝❤ ❛♥❞ ❢♦r♠❛t✐♦♥✳

❚♦ ♠② ❝♦✲✇♦r❦❡rs ❛♥❞ ♦✣❝❡ ♠❛t❡s✱ t❤❛♥❦ ②♦✉ ❢♦r ♣r♦✈✐❞✐♥❣ ❛ ♥✐❝❡ ✇♦r❦✐♥❣ ❡♥✈✐r♦♥✲ ♠❡♥t✱ ❢♦r ❛tt❡♥t✐✈❡ ❡❛rs t♦ ♦❜st❛❝❧❡s ❛♥❞ s♠✐❧② ❢❛❝❡s ❢♦r t❤❡ ❤❛♣♣②✲❤♦✉rs✳ ■t ✇❛s ♠② ♣r✐✈✲ ✐❧❡❣❡ t♦ ✇♦r❦ ✇✐t❤ ❊❞✉❛r❞♦ ❙✐♠õ❡s✱ P❛✉❧♦ ◆✐❣r♦✱ ❋❡r♥❛♥❞♦ ●♦♥ç❛❧✈❡s✱ ▼❛r❝♦ ❙❛♠♣❛✐♦✱ ❍❡♥r✐q✉❡ ●♦♠❡s✱ ▲❡♦♥❛r❞♦ ▲❛❣♦✱ ▼❛r❝❡❧♦ ❚❡✐①❡✐r❛✱ ■❣♦r P✐❡r✐♥✱ ◆❛❞✐❛ ❖t❛✱ ❆❧❢r❡❞♦ ●❛② ◆❡t♦✱ ▼ár❝✐❛ ❙❛t♦✱ ❆❧❡①❛♥❞r❡ ❇❡❧❧❡t✐✱ ▲✉ís ❇✐t❡♥❝♦✉rt✱ ❘✐❝❛r❞♦ ▲❛❤✉❡rt❛ ❛t ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦✱ ❛♥❞ t♦ ❦♥♦✇ s♦♠❡ ♦❢ t❤❡✐r ❞❡❧✐❣❤t❢✉❧ ❢❛♠✐❧✐❡s✳

■ ❛❧s♦ t❤❛♥❦ t❤❡ ♥❡✇ ❢r✐❡♥❞s ■ ♠❛❞❡ ✐♥ ❍❛♥♥♦✈❡r✱ ✇❤♦ ♠❛❞❡ ♠❡ ❢❡❡❧ ✇❡❧❝♦♠❡ s♦ ❢❛r ❢r♦♠ ❤♦♠❡✿ ❆❥❛② ❍❛r✐s❤✱ ▼♦❤❛♠❡❞ ❆♥♥❞✐❢✱ ▼♦❤❛♠♠❛❞r❡③❛ ❍♦❥❥❛t✐✱ ❙❡❜❛st✐❛♥ ❩❡❧❧❡r✱ ❙❛❧❡❤ ❨❛③❞❛♥✐✱ ❱✐♥❤ ❍✐❡♣✱ ❚♦❜✐❛s ❙t❡✐♥❡r ❛♥❞ ❈❤r✐st✐❛♥ ❲❡✐ÿ❡♥❢❡❧s ❛♠♦♥❣ ♦t❤❡rs ❛t ■❑▼✳ ❆❧s♦ t♦ ▼rs✳ ❍❛❧❢❛r ❛♥❞ ▼rs✳ ❙❝❤✉❧t❡ ❢♦r ❛❧❧ t❤❡ ❤❡❧♣✳

■ ✇❛s ❛❧s♦ ❧✉❝❦② t♦ ✜♥❞ ✐♥ ●❡r♠❛♥② ❣r❡❛t ♥❡✇ ❢r✐❡♥❞s ♦✉ts✐❞❡ ♦❢ t❤❡ ✇♦r❦♣❧❛❝❡✳ ■ t❤❛♥❦ ❈❛r❧♦s ▲♦♣❡s✱ ▼❡❧❛♥✐❡ ❚✐t③ ❛♥❞ ❤❡r ♠❛♥② ❢r✐❡♥❞s✱ ■r❡♥❡ ❞✐ P❛❧♠❛✱ ❘♦❜✐♥ ❈♦r❣✐❡r ❛♥❞ ●❛❜r✐❡❧❛ ❙③❛❢❛rs❦❛✱ ❣r❡❛t ✢❛t♠❛t❡s✱ ▼r✳ ❛♥❞ ▼rs✳ ❋r❛♥❦❡✳ ▼② ❜❡st ❛♣♣r❡❝✐❛t✐♦♥ ❘✐t❛ ❈❛str♦✱ ▲ü❞❡r ▼❡②❡r ❛♥❞ t❤❡ ▼❡②❡r ❢❛♠✐❧② ❢♦r s✉❝❤ ❛ ❣r❡❛t ❤♦s♣✐t❛❧✐t② ❛♥❞ ❛❧s♦ t♦ t❤❡ ❧♦✈✐♥❣ ❚♦t③❦❡ ❢❛♠✐❧②✳

▼② ❛❝❦♥♦✇❧❡❞❣♠❡♥t ❛❧s♦ ❣♦ t♦ ❈◆Pq ✭❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❚❡❝✲ ♥♦❧ó❣✐❝♦✮ ❢♦r ❛ P❤❉ s❝❤♦❧❛rs❤✐♣ ❛♥❞ ❈❆P❊❙ ✭❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✮ ❢♦r s✉♣♣♦rt✐♥❣ ♠② ❡①❝❤❛♥❣❡ st❛② ❛t ■❑▼ ✇✐t❤ ❣r❛♥t ❇❊❳ ✶✹✻✶✽✴✶✸✲✸✳

❆❣r❛❞❡❝✐♠❡♥t♦s

➱ ❝♦♠ ✉♠ ✐♠❡♥s♦ ♣r❛③❡r q✉❡ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❝❛♠✐♥❤❛r❛♠ ❥✉♥t♦ ❝♦♠✐❣♦ ♥❡ss❛ ❥♦r♥❛❞❛✳ ❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s q✉❡ ♠❡ ❞á✱ ♣❡❧❛ ♠✐♥❤❛ ♣❛③ ❡ ♣❡❧❛s ♣❡ss♦❛s q✉❡ ❝♦❧♦❝❛ ❥✉♥t♦ ❛ ♠✐♠✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r ♠❡ ❞❛r s❡❣✉r❛♥ç❛ q✉❡ ♦ ❛♠♦r é ❣r❛♥❞❡ ❡ ♥ã♦ ❝♦♥❤❡❝❡ ❞✐stâ♥❝✐❛s✳ ❆ ♠✐♥❤❛s ❛✈ós ■③❛ ❡ ■♥ês ♣♦r t♦❞♦ ❝❛r✐♥❤♦ ❡ ♣♦r s❡♠♣r❡ r❡③❛r❡♠ ♣♦r ♠✐♠✱ ❛s ❛✈ós ✏❡♠♣r❡st❛❞❛s✑ ❊✉♥✐❝❡ ❡ ◆❛③✐♥❤❛ ♣♦r ❡♥❣r♦ss❛r❡♠ ❡ss❡ ❝♦r♦✱ ❛ ♠❡✉s ❛✈ôs ❏♦❛❧❞♦ ❡ ▲❡❛❧❞♦ ✭❛♠❜♦s ✐♥ ♠❡♠♦r✐❛♥✮ ♣❡❧❛s r❡❝♦r❞❛ç♦❡s ❡ ❧✐çõ❡s ❡ t❛♠❜é♠ ❛♦ ❛✈ô ✏❡♠♣r❡st❛❞♦✑ ❖s✈❛❧❞♦ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ❛s ❝♦♥✈❡rs❛s s❡♠♣r❡ ✐♥t❡r❡ss❛♥t❡s✳

❉❡❞✐❝♦ ♠✉✐t♦ ❝❛r✐♥❤♦ ❛ ♠❡✉s ❚✐♦s ❡ ❚✐❛s q✉❡ ♠❡ ❣✉✐❛♠ ❡ s✉♣♦rt❛♠ s❡♠♣r❡✳ ❊♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛s s❡♠♣r❡ ♣r❡s❡♥t❡s t✐❛s✲♠ã❡s ❚❡r❡③❛ ❡ ▼❛r❧②✱ ❘♦s❛ ❡ ➶♥❣❡❧❛❀ ❡ t✐❛ ❱â♥✐❛✱ ❝♦♠♣❛♥❤❡✐r❛ ❞❡ ❞♦✉t♦r❛❞♦✱ q✉❡ ♠❡ ✐♥❝❡♥t✐✈❛ ❡ ❣✉✐❛ ♣❛r❛ ❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✱ ♦❜r✐❣❛❞♦ ♣❡❧❛ ❝♦♥✜❛♥ç❛✳ ❆ ♠❡✉s t✐♦s ❏♦ã♦✱ ❏♦❝❛✱ ❇ôs❝♦✱ ❘✐❝❛r❞♦ ❡ ■✈♦ ♣♦r s❡r❡♠ ❣r❛♥❞❡s ❡①❡♠♣❧♦s✳ ❆ ♠❡✉s ♣r✐♠♦s✲✐r♠ã♦s ❏♦❛❧❞♦✱ ❊✈❡❧②♥❡ ❡ ■③✐♥❤❛ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣❛♥❤✐❛ ❡ ❛♠✐③❛❞❡ ✈❡r❞❛❞❡✐r❛✳ ❆ ♠❡✉s ♣r✐♠♦s✲❛✜❧❤❛❞♦s ❱ít♦r✱ ▼❛t❤❡✉s✱ ❘♦❞r✐❣♦✱ ❊❞✉❛r❞♦✱ ●❡♦✈❛♥❛ ❡ ❚❤❛✐♥á✱ q✉❡ ❥á ❝r❡s❝❡r❛♠✱ ❡♥❥♦❛r❛♠ ❞❛s ❜r✐♥❝❛❞❡✐r❛s s❡♠ ❣r❛ç❛ ❡ s❡ t♦r♥❛r❛♠ ❝♦♠♣❛♥❤❡✐r♦s✳ ❆ ❏❛✐❛♥❡ ❡ ▲é♦✱ ♠❡✉s ♥♦✈♦s ♣r✐♠♦s✱ ❡ ❛ ❏ú❧✐❛ ♣♦r s❡✉ ❝❛r✐♥❤♦ s✐♥❝❡r♦✳

❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❞❡ ❉♦r❡s ♣❡❧❛ ❛t❡♥çã♦ ❡ ❛❢❡t♦✱ ❡♠ ❡s♣❝✐❛❧ ❛ ❚❤✐❛❣♦ ♣❡❧❛ t♦r❝✐❞❛ ❡ ❛♣♦✐♦✳ ❚✉❞♦ q✉❡ ♣✉❞❡r ❢❛③❡r ♣♦r ♠✐♥❤❛ ❢❛♠✐❧✐❛ é ♣♦✉❝♦ ❞✐❛♥t❡ ❞♦ q✉❡ r❡♣r❡s❡♥t❛ ❛ ♠✐♠✳

❆❣r❛❞❡ç♦ ❛♦s ❛♠✐❣♦s q✉❡ ❡s♣❛❧❤❡✐ ♣♦r ❛í ❡ q✉❡ ♠❡ ❢❛③❡♠ s❡♥t✐r ❜❡♠ ❡♠ t♦❞♦ ❧✉❣❛r✳ ❆ ❆❧❡①❛♥❞r❡✱ ❆♥♥❡✱ ❲❛❧t❡r ❡ ❮r✐s ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ♥♦ ❞✐❛ ❛ ❞✐❛ ❞❡ q✉❡♠ ♠♦r❛ ❥✉♥t♦✳ ❆ ❘②❛♥✱ ▼♦❛❝✐r✱ ❈❛r❧✐♥❤♦s✱ ❨✉r✐✱ ❍❡❧♦✐s❛✱ ❋❛❜✐❛♥❛✱ ❘♦❞r✐❣♦ ❈♦❝❛t❡ ❡ ♦✉tr❛s ✜❣✉r❛s ♣❛✉❧✐st❛♥❛s✳ ❆ ❆r✐❡❧✱ ▼❛r❝❡❧✐♥❤♦✱ ❆❧♦ís✐♦✱ ❙❛♠✉❡❧✱ ❍✐❣♦r ❙ér❣✐♦✱ ▼❛r❝❡❧✱ ❆♥ís✐❛✱ ❚❤❛ís❛✱ ●✉st❛✈♦✱ ❏✉✲ ❧✐❛♥❛✱ ❘♦♥②✱ ❏♦r❣❡✱ ❋á❜✐♦✱ ▲✉❝❛s✱ ❉✐♦❣♦✱ ●✉❣✉✱ ▼❛❞s♦♥✱ ❋❧á✈✐♦✱ P❧í♥✐♦ ❡ ❮❝❛r♦✳ ❆ ❈❛r❧ã♦ ❍❡r♠í♥✐♦✱ ❱ít♦r ❡ ♦s ♦✉tr♦s ✸✵ ❡ t❛♥t♦s ♣r✐♠♦s ❡ t✐♦s✳

P♦r ✜♠✱ ♠❡✉ ♠❛✐♦r ❛❣r❛❞❡❝✐♠❡♥t♦ ❛ ♠❡✉s ♣❛✐s✿ ❏♦r❣❡ ▲✐♠❛ ❡ ▲✉③✐❛✳ ❖❜r✐❣❛❞♦ ♣❡❧♦ s✉♣♦rt❡✱ ❛♣♦✐♦✱ t♦r❝✐❞❛✱ ❝❛r✐♥❤♦ ❡ t♦❞❛s ❛s ❢♦r♠❛s ❞❡ ❜❡♠ q✉❡ s❡♠♣r❡ ❞✐r✐❣✐r❛♠ ❛ ♠✐♠✳ ❉❡❞✐❝♦ ❡ss❡ tr❛❜❛❧❤♦ ❛ ✈♦❝ês✳

❘❡s✉♠♦

❈♦st❛✱ ❏✳ ❈✳ ✳ ❈♦♥❞✐çõ❡s ❡ss❡♥❝✐❛✐s ❞❡ ❝♦♥t♦r♥♦ ❡ ✐♥t❡r❢❛❝❡ ♥❛ ❛♥á❧✐s❡ ❞❡ ❝❛s✲ ❝❛s ❝♦♠ ♠ét♦❞♦s s❡♠ ♠❛❧❤❛✳ ✷✵✶✺✳ ✻✸ ❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳

▼ét♦❞♦s s❡♠ ♠❛❧❤❛ ❣❡r❛♠ ❝❛♠♣♦s ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ ❛❧t❛ ❝♦♥t✐♥✉✐❞❛❞❡✱ ❝♦♥✈❡✲ ♥✐❡♥t❡s ♣❛r❛ ❡str✉t✉r❛s ✜♥❛s ❝♦♠♦ ❝❛s❝❛s✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❛✉sê♥❝✐❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❉❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r ❞✐✜❝✉❧t❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❝♦♥❞✐çõ❡s ❡ss❡♥❝✐❛✐s ❞❡ ❝♦♥t♦r♥♦✱ ❥á q✉❡ ♦s ❝❛♠♣♦s ❞❡ ❛♣r♦①✐♠❛çã♦ ❡ t❡st❡ ♥ã♦ ♣♦❞❡♠ s❡r ♠♦❧❞❛❞♦s ❛♦s ✈❛❧♦r❡s ❞❡ ❝♦♥t♦r♥♦✳ ❯♠ ♣r♦❜❧❡♠❛ s✐♠✐❧❛r ❛♣❛r❡❝❡ q✉❛♥❞♦ ❞✐❢❡r❡♥t❡s r❡❣✐õ❡s ❞❡ ❛♣r♦①✐♠❛çã♦ ♣r❡❝✐s❛♠ s❡r ❥✉♥t❛❞❛s ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ♠✉❧t✐✲r❡❣✐õ❡s ❝♦♠♦ ❞♦❜r❛s✱ ✈✐♥❝♦s ♦✉ ❥✉♥çõ❡s✳ ❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ três ♠ét♦✲ ❞♦s ❞❡ ✐♠♣♦s✐çã♦ ❛♠❜❛s ❝♦♥❞✐çõ❡s ❝✐♥❡♠át✐❝❛s✿ ♦ ❥á ❝♦♥❤❡❝✐❞♦ ♠ét♦❞♦ ❞♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✱ ✉s❛❞♦ ❞❡s❞❡ ♦ ❝♦♠❡ç♦ ❞♦ ♠ét♦❞♦ ❞❡ ●❛❧❡❦✐♥ s❡♠ ❡❧❡♠❡♥t♦s ✭❊❋●✮❀ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ♣✉r❛❀ ❡ ♦ r❡❝❡♥t❡♠❡♥t❡ r❡❞❡s❝♦❜❡rt♦ ♠ét♦❞♦ ❞❡ ◆✐ts❝❤❡✳ ◆ós ✉s❛♠♦s ❛ té❝♥✐❝❛ ❞❡ ❞✐s❝r❡t✐③❛çã♦ ❝♦♠ ❊❋● ♣❛r❛ ❝❛s❝❛s ❡s♣❡ss❛s ❞❡ ❘❡✐ss♥❡r✲▼✐♥❞❧✐♥ ❡ ❛❞❛♣t❛♠♦s ❛ ❢♦r♠❛ ❢r❛❝❛ ❞❡ ❢♦r♠❛ ❛ s❡♣❛r❛r ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❡ r♦t❛çã♦ ❡ ♦❜t❡r ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡st❛❜✐❧✐③❛çã♦ ❞✐❢❡r❡♥t❡s ❡ ❛♣r♦♣r✐❛❞♦s✳ ❊ss❛ ❛❜♦r❞❛❣❡♠ ♣❡r♠✐t❡ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❝❛s❝❛s ❞✐s❝♦♥tí♥✉❛s ❡ ♦ r❡✜♥❛♠❡♥t♦ ❧♦❝❛❧ ❡♠ ♣r♦❜❧❡♠❛s ♠✉❧t✐✲r❡❣✐õ❡s✳ P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❝❛s❝❛s✱ ♠ét♦❞♦s s❡♠ ♠❛❧❤❛✱ ♠ét♦❞♦ ❞❡ ◆✐ts❝❤❡✱ ♣r♦❜❧❡♠❛s ♠✉❧t✐✲r❡❣✐õ❡s✱ ❝❛s❝❛s ❝♦♠ ❝✐s❛❧❤❛♠❡♥t♦✳

❆❜str❛❝t

❈♦st❛✱ ❏✳ ❈✳ ✳ ❊ss❡♥t✐❛❧ ❜♦✉♥❞❛r② ❛♥❞ ✐♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ♠❡s❤❧❡ss ❛♥❛❧✲ ②s✐s ♦❢ s❤❡❧❧s✳ ✷✵✶✺✳ ✻✸ ♣✳ ❚❤❡s✐s ✭❉♦❝t♦r❛t❡✮ ✲ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳

▼❡s❤❧❡ss ♠❡t❤♦❞s ♣r♦✈✐❞❡ ❛ ❤✐❣❤❧② ❝♦♥t✐♥✉♦✉s ❛♣♣r♦①✐♠❛t✐♦♥ ✜❡❧❞✱ ❝♦♥✈❡♥✐❡♥t ❢♦r t❤✐♥ str✉❝t✉r❡s ❧✐❦❡ s❤❡❧❧s✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ❧❛❝❦ ♦❢ ❑r♦♥❡❝❦❡r ❉❡❧t❛ ♣r♦♣❡rt② ♠❛❦❡s t❤❡ ❢♦r✲ ♠✉❧❛t✐♦♥ ♦❢ ❡ss❡♥t✐❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♥♦t str❛✐❣❤t❢♦r✇❛r❞✱ ❛s t❤❡ tr✐❛❧ ❛♥❞ t❡st ✜❡❧❞s ❝❛♥♥♦t ❜❡ t❛✐❧♦r❡❞ t♦ ❜♦✉♥❞❛r② ✈❛❧✉❡s✳ ❙✐♠✐❧❛r ♣r♦❜❧❡♠ ❛r✐s❡ ✇❤❡♥ ❞✐✛❡r❡♥t ❛♣♣r♦①✐♠❛✲ t✐♦♥ r❡❣✐♦♥s ♠✉st ❜❡ ❥♦✐♥❡❞✱ ✐♥ ❛ ♠✉❧t✐✲r❡❣✐♦♥ ♣r♦❜❧❡♠✱ s✉❝❤ ❛s ❦✐♥❦s✱ ❢♦❧❞s ♦r ❥♦✐♥ts✳ ❚❤✐s ✇♦r❦ ♣r❡s❡♥ts t❤r❡❡ ❛♣♣r♦❛❝❤❡s t♦ ✐♠♣♦s❡ ❜♦t❤ ❦✐♥❡♠❛t✐❝ ❝♦♥❞✐t✐♦♥s✿ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ▲❛✲ ❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡r ♠❡t❤♦❞✱ ✉s❡❞ s✐♥❝❡ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❊❧❡♠❡♥t ❋r❡❡ ●❛❧❡r❦✐♥ ♠❡t❤♦❞❀ ❛ ♣✉r❡ ♣❡♥❛❧t② ❛♣♣r♦❛❝❤❀ ❛♥❞ t❤❡ r❡❝❡♥t❧② r❡❞✐s❝♦✈❡r❡❞ ❛❧t❡r♥❛t✐✈❡ ♦❢ ◆✐ts❝❤❡✬s ▼❡t❤♦❞✳ ❲❡ ✉s❡ t❤❡ ❊❋● ❞✐s❝r❡t✐③❛t✐♦♥ t❡❝❤♥✐q✉❡ ❢♦r t❤✐❝❦ ❘❡✐ss♥❡r✲▼✐♥❞❧✐♥ s❤❡❧❧s ❛♥❞ ❛❞❛♣t t❤❡ ✇❡❛❦ ❢♦r♠ ❛s t♦ s❡♣❛r❛t❡ ❞✐s♣❧❛❝❡♠❡♥t ❛♥❞ r♦t❛t✐♦♥❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❛♥❞ ♦❜t❛✐♥ s✉✐t❛❜❧❡ ❛♥❞ s❡♣❛r❛t❡ st❛❜✐❧✐③❛t✐♦♥ ♣❛r❛♠❡t❡rs✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❡♥❛❜❧❡s t❤❡ ♠♦❞❡❧✐♥❣ ♦❢ ❞✐s❝♦♥t✐♥✉♦✉s s❤❡❧❧s ❛♥❞ ❧♦❝❛❧ r❡✜♥❡♠❡♥t ♦♥ ♠✉❧t✐✲r❡❣✐♦♥ ♣r♦❜❧❡♠s✳

❑❡②✇♦r❞s✿ s❤❡❧❧s✱ ♠❡s❤❧❡ss ♠❡t❤♦❞s✱ ♥✐ts❝❤❡✬s ♠❡t❤♦❞✱ ♠✉❧t✐✲r❡❣✐♦♥ ♣r♦❜❧❡♠s✱ s❤❡❛r ❞❡✲ ❢♦r♠❛❜❧❡ s❤❡❧❧s✳

❈♦♥t❡♥ts

✶ ■♥tr♦❞✉❝t✐♦♥ ✶

✷ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✹

✷✳✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❑✐♥❡♠❛t✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✸ ❙t❛t✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✺ ▼❛t❡r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✻ ■♥✐t✐❛❧ ❉❡❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✼ ❲❡❛❦ ❋♦r♠ ♦❢ ❊q✉✐❧✐❜r✐✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✽ ▼✉❧t✐✲r❡❣✐♦♥ s❤❡❧❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✸ ▼❡s❤❧❡ss ❆♣♣r♦①✐♠❛t✐♦♥ ✶✻

✹ ❑✐♥❡♠❛t✐❝ ❈♦♥str❛✐♥ts ✷✵

✹✳✶ ▲❛❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✷ P❡♥❛❧t② ▼❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✸ ◆✐ts❝❤❡✬s ▼❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✺ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ✹✶

✺✳✶ ❙t❛❜✐❧✐③❛t✐♦♥ ❢♦r ❊❇❈s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✶✳✶ ❉✐s♣❧❛❝❡♠❡♥t ❉❖❋s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✶✳✷ ❘♦t❛t✐♦♥ ❉❖❋s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✷ ❙t❛❜✐❧✐③❛t✐♦♥ ❢♦r ■♥t❡r❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✷✳✶ ❉✐s♣❧❛❝❡♠❡♥t ❉❖❋s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✷✳✷ ❘♦t❛t✐♦♥ ❉❖❋s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✸ P✐♥❝❤❡❞ ❈②❧✐♥❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✺✳✹ ❚✲❜❡❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✻ ❈♦♥❝❧✉s✐♦♥s ✺✻

❈❤❛♣t❡r ✶

■♥tr♦❞✉❝t✐♦♥

▼❡s❤❧❡ss ❛♣♣r♦①✐♠❛t✐♦♥s ♣r♦✈✐❞❡ ❛ ❣r❡❛t r❡s♦✉r❝❡ ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ str✉❝t✉r❡s ❛s ❤✐❣❤ ♦r❞❡r ❝♦♥t✐♥✉✐t② ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t❡❞ ✜❡❧❞s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞✳ ❚❤✐s ❢❡❛t✉r❡ ✐s ✇❡❧❧ s✉✐t❡❞ ❢♦r t❤✐♥ s②st❡♠s ❧✐❦❡ s❤❡❧❧s✱ ♣r♦✈✐❞❡❞ t❤❛t t❤❡ ❣❡♦♠❡tr② ❝❛♥ ❜❡ ❜❡tt❡r ❛♣♣r♦①✐♠❛t❡❞ ❛♥❞ str❡ss❡s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s s♠♦♦t❤❧② ❛s ❞❡s✐r❡❞✳ ❚❤❡② ❤❛✈❡ ❜❡❡♥ ❛♣♣❧✐❡❞ t♦ t❤✐♥ ❑✐r❝❤❤♦✛✲▲♦✈❡ s❤❡❧❧s ❬✸✼✱ ✸✶❪✱ ✇❤❡r❡C1_{❝♦♥t✐♥✉✐t② ✐s ♥❡❝❡ss❛r②✱ ♠♦❞❡r❛t❡❧② t❤✐❝❦ ❘❡✐ss♥❡r✲} ▼✐♥❞❧✐♥ s❤❡❧❧s ❬✹✽✱ ✺✽✱ ✶✷❪ ❛♥❞ s♦❧✐❞✲❧✐❦❡ s❤❡❧❧s ❜❛s❡❞ ♦♥ ✸❉ ❝♦♥t✐♥✉✉♠ ❬✸✽❪✳ ❍♦✇❡✈❡r✱ t❤❡ ♥♦♥✲✐♥t❡r♣♦❧❛t♦r② ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ s✉❝❤ ❛♣♣r♦①✐♠❛♥ts ♠❛❦❡s t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❡ss❡♥t✐❛❧ ❜♦✉♥❞❛r② ❛♥❞ ✐♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥s ♥♦t str❛✐❣❤t❢♦r✇❛r❞✳

❲❡ ✐♥tr♦❞✉❝❡ ✐♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❝♦♥t✐♥✉✐t② ❛♥❞ ❡q✉✐❧✐❜r✐✉♠ ❛❝r♦ss ❛ ❜♦✉♥❞❛r② ❜❡t✇❡❡♥ t✇♦ ❞✐s❝r❡t✐③❛t✐♦♥ ❞♦♠❛✐♥s✳ ❚❤❡ ❝♦✉♣❧✐♥❣ ♦❢ ♠✉❧t✐♣❧❡ r❡❣✐♦♥s ✐s q✉✐t❡ str❛✐❣❤t✲ ❢♦r✇❛r❞ ✐♥ ❋✐♥✐t❡ ❊❧❡♠❡♥ts ✭❋❊✮ ✇✐t❤ ♠❛t❝❤✐♥❣ ♥♦❞❡s ❛❧♦♥❣ t❤❡ ✐♥t❡r❢❛❝❡✱ ❜✉t t❤✐s ❧✐♠✐ts t❤❡ ❧✐❜❡rt② ♦❢ ❡❛❝❤ r❡❣✐♦♥✬s r❡✜♥❡♠❡♥t✳ ❆ ❣r♦✇✐♥❣ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❈♦♠♣✉t❡r ❆✐❞❡❞ ❊♥✲ ❣✐♥❡❡r✐♥❣ ✐s t❤❡ ♣♦ss✐❜✐❧✐t② t♦ ❛♥❛❧②③❡ ❢✉❧❧✱ ❝♦♠♣❧❡① ♠♦❞❡❧s ❝♦♠♣♦s❡❞ ♦❢ ♠✉❧t✐♣❧❡ ♣❛rts✱ ❦❡❡♣✐♥❣ t❤❡ ♣♦ss✐❜✐❧✐t② t♦ r❡✜♥❡ t❤❡ r❡s♣♦♥s❡ ✐♥ ♠♦st ❝r✐t✐❝❛❧ ❛r❡❛s✳ ■♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡❝❤❛♥✐❝s✱ t❤✐s tr❛♥s❧❛t❡s ✐♥t♦ t❤❡ ♥❡❡❞ ❢♦r ♠✉❧t✐♣❧❡ r❡❣✐♦♥s t♦ ❜❡ ❞❡s❝r✐❜❡ t♦❣❡t❤❡r ✇❤✐❧❡ ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❛❜✐❧✐t② t♦ ❜❡ r❡✜♥❡❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳ ❖♥❡ r❡❣✐♦♥ ♠✐❣❤t ♣♦ss✐❜❧② ♥❡❡❞ ❛ ♠♦r❡ r❡✜♥❡❞ s♦❧✉t✐♦♥✱ ♦r ❞✐✛❡r❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ t❡❝❤♥✐q✉❡s ♠✐❣❤t ❜❡ ❡♠♣❧♦②❡❞ ✐♥ ❡❛❝❤ s✐❞❡ ♦❢ t❤❡ ✐♥t❡r❢❛❝❡ ✭✐✳❡✳✱ ❋❊s ❛♥❞ ♠❡s❤❧❡ss✮✳ ❙✐♠✐❧❛r ♣r♦❜❧❡♠ ❛r✐s❡ ✇❤❡♥ ❞❡❛❧✐♥❣ ✇✐t❤ ❛♣♣r♦①✐♠❛t✐♦♥s t❤❛t ❞♦ ♥♦t ❢✉❧✜❧❧ t❤❡ ❑r♦♥❡❝❦❡r ❞❡❧t❛ ♣r♦♣❡rt② s✉❝❤ ❛s ♠♦st ♠❡s❤❧❡ss ♠❡t❤♦❞s ✭♠❡s❤❧❡ss ❛♣♣r♦①✐♠❛t✐♦♥s ♦♥ ❜♦t❤ ❞♦♠❛✐♥s✮✳ ❋♦r ♠❛♣♣❡❞ s❤❡❧❧ t❤❡♦r✐❡s✱ ❡❛❝❤ ❞♦♠❛✐♥ ♠✉st ❛❧s♦ ❜❡ s♠♦♦t❤✱ s♦ ❛♥② ❦✐♥❦ ♦r ❥✉♥❝t✐♦♥ ♦❢ s❤❡❧❧s ✐s tr❡❛t❡❞ ❛s ❛ ✐♥t❡r❢❛❝❡ ❜❡t✇❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ❛♣♣r♦①✐♠❛t❡❞ ✜❡❧❞s✳

❊ss❡♥t✐❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭❊❇❈s✮ ❛♥❞ ✐♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥s ✭■❈s✮ ❛r❡ ❢♦r♠✉❧❛t❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛♥❞ ❤❛✈❡ ❜❡❡♥ ❞✐s❝✉ss❡❞ s✐♥❝❡ t❤❡ ❡❛r❧② ❞❡✈❡❧♦♣♠❡♥t ♦❢ ♠❡s❤❧❡ss ♠❡t❤♦❞s✳ ❋r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣✱ ❛s ✐♥ t❤❡ ❝❧❛ss✐❝❛❧ ❊❧❡♠❡♥t✲❋r❡❡ ●❛❧❡r❦✐♥ ▼❡t❤♦❞ ✭❊❋●✱ ❬✻❪✮✱ ▲❛✲ ❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡rs ❤❛✈❡ ❜❡❡♥ ✉s❡❞ t♦ ❡♥❢♦r❝❡ ❊❇❈s✱ ❜✉t ❞✐s❛❞✈❛♥t❛❣❡s ❧✐❦❡ ✐♥❝r❡❛s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✉♥❦♥♦✇♥s ❢♦r t❤❡ ♣r♦❜❧❡♠ ❛♥❞ ❧✐♠✐t❛t✐♦♥s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ✜❡❧❞s ❝♦♠♣❡❧❧❡❞ ♠♦r❡ r❡s❡❛r❝❤ ♦♥ t❤❡ s✉❜❥❡❝t ❬✸✺✱ ✶✽❪✳ ❙♦❧✉t✐♦♥s r❛♥❣❡❞ ❢r♦♠ ❝♦✉♣❧✐♥❣ ✇✐t❤ ❋❊s

✶✳ ■♥tr♦❞✉❝t✐♦♥ ✷ ❛❧♦♥❣ t❤❡ ❜♦✉♥❞❛r✐❡s ❬✼✱ ✷✾❪ t♦ t❤❡ ✉s❡ ♦❢ t❤❡ ❞♦♠❛✐♥ s❤❛♣❡ ❢✉♥❝t✐♦♥s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ❬✹✷✱ ✹✸✱ ✾❪✳

❘❡❝❡♥t❧②✱ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❤❛s ❜❡❡♥ r❡✈✐s✐t❡❞✿ t❤❡ ✐♥t❡r✐♦r ♣❡♥❛❧t② ♠❡t❤♦❞✱ ✉s✉❛❧❧② r❡✲ ❢❡rr❡❞ t♦ ❛s ◆✐ts❝❤❡✬s ▼❡t❤♦❞ ❬✹✼❪✱ ✇❤✐❝❤ ✐❞❡♥t✐✜❡s t❤❡ ▲❛❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡rs ✇✐t❤ t❤❡ ✢✉① ❛t t❤❡ ❡ss❡♥t✐❛❧ ♦r ✐♥t❡r❢❛❝❡ ❜♦✉♥❞❛r② ❛♥❞ ✐♥tr♦❞✉❝❡s ❛ ♣❡♥❛❧t②✲❧✐❦❡ st❛❜✐❧✐③❛t✐♦♥ ♣❛r❛♠✲ ❡t❡r✱ ✇❤✐❝❤ ❣✉❛r❛♥t❡❡s t❤❡ ❝♦❡r❝✐✈✐t② ♦❢ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠✳

❆♠♦♥❣ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✱ t❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ✉s❡❞ t♦ ✐♠♣♦s❡ ❊❇❈s ✐♥ P♦✐ss♦♥✬s ♣r♦❜❧❡♠s ✇✐t❤ ♠❡s❤❧❡ss ♠❡t❤♦❞s ❬✷✵✱ ✶✽❪ ❛♥❞ ❋❊s ❬✺✼❪✳ ■t ❤❛s ❜❡❡♥ ❛♣♣❧✐❡❞ ✇✐t❤ ❡❳t❡♥❞❡❞ ❋✐♥✐t❡ ❊❧❡♠❡♥ts ✭❳❋❊▼✮ ❢♦r ❊❇❈s ❛♥❞ ■❈s ✐♥ s✐♥❣❧❡✲✜❡❧❞ ♣r♦❜❧❡♠s ❬✶✺✱ ✷✺✱ ✷✽❪✱ t♦ ❡♥❢♦r❝❡ ✇❡❛❦ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❜❡t✇❡❡♥ ❡❧❛st✐❝ ♠❛t❡r✐❛❧s ❬✺✶✱ ✺✷✱ ✷✱ ✶❪❀ ❛♥❞ ❛♣♣❧✐❡❞ t♦ ♥♦♥ ❝♦♥❢♦r♠✐♥❣ ❡♠❜❡❞❞❡❞ ❋❊ ♠❡s❤❡s ❢♦r ❡❧❛st♦st❛t✐❝s ❬✺✸✱ ✺✹❪✳ ❇✲s♣❧✐♥❡s ❛♥❞ ◆❯❘❇s✲❜❛s❡ ❋❊s s❤❛r❡ s✐♠✐❧❛r ❢❡❛t✉r❡s ✇✐t❤ ♠❡s❤❧❡ss ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❝❛♥ ❛❧s♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ◆✐ts❝❤❡✬s ▼❡t❤♦❞✱ ❛s ✐♥ ❬✶✻❪ ❢♦r ✷♥❞ ❛♥❞ ✹t❤ ♦r❞❡r ♣r♦❜❧❡♠s✱ ❬✹✻✱ ✸✸✱ ✹✾✱ ✺✵❪ ❢♦r ❡❧❛st✐❝✐t②✱ ❬✷✼✱ ✷✻❪ ❢♦r t❤✐♥ ♣❧❛t❡s ❛♥❞ ❬✷✷✱ ✷✸❪ ❢♦r t❤✐♥ s❤❡❧❧s✳ ❚❤❡ t❤❡♦r② ❤❛s ❛❧s♦ ❜❡❡♥ ❡①t❡♥❞❡❞ ❢♦r ❝♦♥t❛❝t ✇✐t❤ ❋❊s ❬✺✾❪ ❛♥❞ ❳❋❊▼ ❬✹✱ ✸❪✳

■t ✐s t❤❡ ❛✉t❤♦rs✬ ♦♣✐♥✐♦♥ t❤❛t ✐♥ t❤❡ ❢✉t✉r❡ ◆✐ts❝❤❡✬s ♠❡t❤♦❞ ♠❛② ❜❡❝♦♠❡ t❤❡ st❛♥❞❛r❞ tr❡❛t♠❡♥t t♦ ❡♥❢♦r❝✐♥❣ ❡ss❡♥t✐❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❢♦r ♠❡s❤❢r❡❡ ●❛❧❡r❦✐♥ ♠❡t❤♦❞s✳ ❬✸✾✱ ♣✳ ✶✶✹❪

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ❞❡✈❡❧♦♣s ◆✐ts❝❤❡✬s ♠❡t❤♦❞ ❢♦r t❤❡ ❧✐♥❡❛r ❡❧❛st✐❝ ❛♥❛❧②s✐s ♦❢ s❤❡❛r ❞❡✲ ❢♦r♠❛❜❧❡ ❘❡✐ss♥❡r✲▼✐♥❞❧✐♥ s❤❡❧❧s ❬✶✷❪✱ ❜♦t❤ ❢♦r t❤❡ ✐♠♣♦s✐t✐♦♥ ♦❢ ❜♦✉♥❞❛r② ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ ❢♦r ♠✉❧t✐✲r❡❣✐♦♥ ♣r♦❜❧❡♠s✳ ■♥ t❤❡ ❧❛tt❡r✱ r❡✜♥❡♠❡♥t ♦✈❡r ♦♥❡ ♣♦rt✐♦♥ ♦❢ t❤❡ ❞♦♠❛✐♥ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ✇✐t❤♦✉t ❛✛❡❝t✐♥❣ ♦t❤❡r r❡❣✐♦♥s✱ ❡✈❡♥ ♠❛✐♥t❛✐♥✐♥❣ t❤❡✐r st✐✛♥❡ss ♠❛tr✐✲ ❝❡s✱ ❛♥❞ ❞✐✛❡r❡♥t r❡❣✐♦♥s ❝❛♥ ❜❡ ❞✐s❝r❡t✐③❡❞ ✇✐t❤ ❞✐✛❡r❡♥t ❛♣♣r♦①✐♠❛♥ts✱ ❡✳❣✳✱ ❋❊✲❊❋● ♦r ❊❋●✲■●❆ ❝♦✉♣❧✐♥❣✳

■♥ ❜♦t❤ ❝❛s❡s✱ ✐ts ❛❞✈❛♥t❛❣❡ ♦✈❡r ▲❛❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡rs ✐s t❤❛t ✐t ♣r❡❝❧✉❞❡s t❤❡ ❞❡✜♥✐✲ t✐♦♥ ♦❢ ❛ ♥❡✇ ❛♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡✱ ♥❡✇ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❛♥❞ ❛ ❤②❜r✐❞ ✐♥t❡r♣♦❧❛t✐♦♥✱ ❛❧♦♥❣ ✇✐t❤ ✐ts ❝♦♠♣❧✐❝❛t✐♦♥s✳ ❆ ♣❡♥❛❧t②✲❧✐❦❡ st❛❜✐❧✐③❛t✐♦♥ t❡r♠ ✐s ♥❡❝❡ss❛r② t♦ ❡♥s✉r❡ t❤❡ ❝♦❡r❝✐✈✐t② ♦❢ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ✇❡❛❦ ❢♦r♠ r❡♠❛✐♥s ❝♦♥s✐st❡♥t ❛♥❞✱ ❝♦♥✲ tr❛r② t♦ ❛ ♣✉r❡ ♣❡♥❛❧t② ✐♠♣♦s✐t✐♦♥✱ ❛❝❝✉r❛❝② ✐s ❣✉❛r❛♥t❡❡❞ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ✈❛❧✉❡s ❢♦r t❤❡ ♣❛r❛♠❡t❡r✳

✶✳ ■♥tr♦❞✉❝t✐♦♥ ✸

❚❤r♦✉❣❤♦✉t t❤❡ t❡①t✱ ✐t❛❧✐❝ ●r❡❡❦ ♦r ▲❛t✐♥ ❧♦✇❡r❝❛s❡ ❧❡tt❡rs ✭a✱ b✱ ✳✳✳ ✱ α✱ β✱ ✳✳✳✮ ❞❡✲ ♥♦t❡ s❝❛❧❛r q✉❛♥t✐t✐❡s❀ ❜♦❧❞ ✐t❛❧✐❝ ●r❡❡❦ ♦r ▲❛t✐♥ ❧♦✇❡r❝❛s❡ ❧❡tt❡rs ✭a✱ b✱ ✳✳✳ ✱ α✱ β✱ ✳✳✳✮

❞❡♥♦t❡ ✈❡❝t♦rs ❛♥❞ ❜♦❧❞ ✐t❛❧✐❝ ●r❡❡❦ ♦r ▲❛t✐♥ ❝❛♣✐t❛❧ ❧❡tt❡rs ✭A✱ B✱ ✳✳✳ ✱ Φ✱Γ✱ ✳✳✳✮ ❞❡♥♦t❡

❈❤❛♣t❡r ✷

▲✐♥❡❛r s❤❡❧❧ t❤❡♦r②

✷✳✶ ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ♥♦♥❧✐♥❡❛r s❤❡❧❧ t❤❡♦r② ❞❡✈❡❧♦♣❡❞ ✐♥ ❬✶✹❪ ❤❛s ❜❡❡♥ ❧❛t❡r r❡♠♦❞❡❧❡❞ ✐♥ ❛ ❝♦♥s✐st❡♥t ✇❛② t♦ ❛ ♥❡❛t ❧✐♥❡❛r ❢♦r♠✉❧❛t✐♦♥ ✐♥ ❬✶✷❪ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ r♦t❛t✐♦♥s✳ ❚❤✐s ❝❤❛♣t❡r ♦✉t❧✐♥❡s t❤❡ ♣r♦❝❡ss ❛s ✐♥ ❬✶✵❪ s♦ ❛ ❝♦♥s✐st❡♥t ❧✐♥❡❛r s❤❡❧❧ ♠♦❞❡❧ ✐s ♦❜t❛✐♥❡❞✳ ❚❤❡ s❤❡❧❧ ✐s ♣❛r❛♠❡t❡r✐③❡❞ ❜② ❛ ✢❛t r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ✢❛t ♠✐❞s✉r✲ ❢❛❝❡ ❛♥❞ ❛ t❤✐❝❦♥❡ss h✳ ❚❤❡ ✐♥✐t✐❛❧ ❝✉r✈❡❞ ❝♦♥✜❣✉r❛t✐♦♥ ✐s t❤❡♥ ♦❜t❛✐♥❡❞ ❜② ❛ str❡ss✲❢r❡❡ tr❛♥s❢♦r♠❛t✐♦♥ ❛❢t❡r ✇❤✐❝❤ t❤❡ ❛❝t✉❛❧ ♦r ❡✛❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥ t❛❦❡s ♣❧❛❝❡✳ ❋✐❣✳ ✷✳✶ ❞❡♣✐❝ts t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r t❤❡ ❣❡♦♠❡tr✐❝❛❧❧② ❡①❛❝t s❤❡❧❧✳

▲❡t ei ❜❡ ❛♥ ♦rt❤♦❣♦♥❛❧ ❢r❛♠❡✱ ✇✐t❤ t❤❡ ✈❡❝t♦rs erα ♦♥ t❤❡ r❡❢❡r❡♥❝❡ ♣❧❛♥❡ ❛♥❞ t❤❡ ✈❡❝t♦r er3 ♦rt❤♦❣♦♥❛❧ t♦ ✐t✳ ❆t t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ❛ ♣♦✐♥t ♦❢ t❤❡ s❤❡❧❧ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ✐ts ♣♦s✐t✐♦♥

ξ =ζ +ar, ✭✷✳✶✮

✇❤❡r❡ ζ = ξαer

α ❞❡✜♥❡s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤✐s ♣♦s✐t✐♦♥ ✐♥ t❤❡ ♠✐❞s✉r❢❛❝❡ ❛♥❞ ar = ξ3er3 ✐s t❤❡ t❤r♦✉❣❤✲t❤❡✲t❤✐❝❦♥❡ss ❝♦♠♣♦♥❡♥t✳ ❚❤❡ s❡t [ξ1, ξ2, ξ3] ❞❡✜♥❡s ❛ ❈❛rt❡s✐❛♥ s②st❡♠ ♦❢ ❝♦♦r❞✐♥❛t❡s ✐♥ t❤r❡❡ ❞✐♠❡♥s✐♦♥s✳

❚❤❡ s❤❡❧❧✬s ♠✐❞s✉r❢❛❝❡ ✐s t❤❡ ❞♦♠❛✐♥Ωr _∈er

1×er2 ❛♥❞ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ♠✐❞s✉r❢❛❝❡ ✐s ❛❝t✉❛❧❧② ♦♥ t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ s❤❡❧❧ t❤✐❝❦♥❡ssξ3 ∈[−h_/2,h_/2]

✷✳✷ ❑✐♥❡♠❛t✐❝s

❚❤❡ ✐♥✐t✐❛❧ ❞❡❢♦r♠❛t✐♦♥ ✐s r❡❣❛r❞❡❞ ❛s ❛ ♠❛♣♣✐♥❣ ❢r♦♠ t❤❡ ✢❛t r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥✳ ❚❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ s❤❡❧❧ ♠✐❞s✉r❢❛❝❡ ♣♦✐♥ts ✐s ❞❡s❝r✐❜❡❞ ✐♥ ✐ts ✐♥✐t✐❛❧ str❡ss✲❢r❡❡ ❝♦♥✜❣✉r❛✲ t✐♦♥ ❜②

zo =zo(ξα). ✭✷✳✷✮

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✺

O _{reference configuration}

e

r 3

e

r 1

e

r 2

a r

initial

configuration e

o 1

e

o 2

e

o 3

a

o

current configuration e₁

e2

e₃

a

z o x

o

z x u

❋✐❣✉r❡ ✷✳✶✿ ❙❤❡❧❧ ❦✐♥❡♠❛t✐❝s ❢♦r ❧❛r❣❡ ❞✐s♣❧❛❝❡♠❡♥ts

❚❤❡s❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛r❡ ✜♥✐t❡✱ ❛♥❞ s♦ ❛r❡ t❤❡ r♦t❛t✐♦♥s ♦❢ t❤❡ ♠✐❞s✉r❢❛❝❡ ❛♥❞ t❤❡ ❞✐r❡❝t♦r✳ ◆❡✈❡rt❤❡❧❡ss✱ ❑✐r❝❤❤♦✛✲▲♦✈❡ ❦✐♥❡♠❛t✐❝s ✐s ❛ss✉♠❡❞✱ s♦ t❤❛t t❤❡ ❞✐r❡❝t♦r ao r❡✲

♠❛✐♥s ♥♦r♠❛❧ t♦ t❤❡ ❝✉r✈❡❞ ♠✐❞s✉r❢❛❝❡✳ ❲❡ ❝❛♥ ❞❡s❝r✐❜❡ t❤✐s r♦t❛t✐♦♥ ❜② t❤❡ t❡♥s♦r Qo

s♦

ao =Qoar =Qoξ3er3. ✭✷✳✸✮

❚❤❡ r❡❢❡r❡♥❝❡ ❢r❛♠❡ ❢♦❧❧♦✇s t❤❡ s❛♠❡ r♦t❛t✐♦♥✱

eo_i =Qoer_i, ✭✷✳✹✮

✇❤❛t ♠❛❦❡s ✐t ❞❡s❝r✐❜❛❜❧❡ ❜②

Qo =eo_i ⊗er_i ✭✷✳✺✮

❆s t❤❡ ✐♥✐t✐❛❧ ❢r❛♠❡ eo

i ❢♦❧❧♦✇s t❤❡ ♠✐❞s✉r❢❛❝❡ ❝✉r✈❛t✉r❡✱ ✐t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ ❜② t❤❡ ♠❛♣♣✐♥❣ ❝♦♦r❞✐♥❛t❡s✿

eo1 =

zo ,1

kzo ,1k

, ✭✷✳✻❛✮

eo2 =e o 3 ×e

o

1 ❛♥❞ ✭✷✳✻❜✮

eo3 =

zo ,1×zo,2

kzo

,1×zo,2k

. ✭✷✳✻❝✮

eo

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✻

❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝r♦ss✲s❡❝t✐♦♥❛❧ str❛✐♥ ✈❡❝t♦rs

ηo_α =zo_α−Qoer_α ❛♥❞ ✭✷✳✼❛✮

κo_α =❛①✐❛❧ Qo_,αQoT

,Ko_α =❙❦❡✇(κo_α) ✭✷✳✼❜✮

❛t ❤❛♥❞✱ t❤❡ ✐♥✐t✐❛❧ str❛✐♥ ✈❡❝t♦rs ❛t ❛♥② ♠❛t❡r✐❛❧ ♣♦✐♥t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜②

γo_α =ηo_α+κoα×a o

, ✭✷✳✽✮

s♦ t❤❛t t❤❡ ✐♥✐t✐❛❧ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❜②

Fo = (ηo_α+Ko_αao)⊗e_αr +Qo =γo_α⊗er_α+Qo. ✭✷✳✾✮ ❚❤❡ q✉❛♥t✐t✐❡s ❛❜♦✈❡ ❛r❡ ❛✛❡❝t❡❞ ❜② s✉♣❡r♣♦s❡❞ r✐❣✐❞ ❜♦❞② ♠♦t✐♦♥s ❛♥❞ ♥♦t s✉✐t✲ ❛❜❧❡ ❢♦r ❛♥ ♦❜❥❡❝t✐✈❡ t❤❡♦r②✳ ❚❤❡r❡❢♦r❡✱ ✐t✬s ❝♦♠♣❡❧❧✐♥❣ t♦ ✐♥tr♦❞✉❝❡ t❤❡✐r ❜❛❝❦✲r♦t❛t❡❞ ❝♦✉♥t❡r♣❛rts ❛s ❢♦❧❧♦✇s

γorα =Q oT

γoα =η or α +κ

or α ×a

r _{✭✷✳✶✵❛✮}

ηor_α =QoTηo_α =QoTzo_,α−er_α ✭✷✳✶✵❜✮

κor_α =QoTκo_α = eo2,α·e e 3

er1 + e o 3,α·e

e 1

er2+ e o 1,α·e

e 2

er3 ✭✷✳✶✵❝✮

For =QoTFo ✭✷✳✶✵❞✮

❋♦r ❧❛t❡r ✉s❡✱ ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❣❛t❤❡r t❤❡ ✐♥✐t✐❛❧ ❝r♦ss✲s❡❝t✐♦♥❛❧ str❛✐♥ ✈❡❝t♦rs ✐♥ t✇♦ ❣❡♥❡r❛❧✐③❡❞ ✐♥✐t✐❛❧ str❛✐♥ ✈❡❝t♦rs

εor_α = "

ηor α

κor α

#

. ✭✷✳✶✶✮

◆❡①t✱ t❤❡ ❛❝t✉❛❧ ♠♦t✐♦♥ ♦❢ t❤❡ s❤❡❧❧ t❛❦❡s ♣❧❛❝❡✳ ❚❤❡ ❞✐s♣❧❛❝❡♠❡♥t ♦❢ ❛♥② ♣♦✐♥t ✐♥ t❤❡ s❤❡❧❧ ❝❛♥ ❜❡ ♦♥❝❡ ❛❣❛✐♥ s♣❧✐t ✐♥t♦ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t u ♦❢ ✐ts ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ ♠✐❞s✉r❢❛❝❡

❛♥❞ ❛ r♦t❛t✐♦♥ ♦❢ t❤❡ ❞✐r❡❝t♦r a =Qeao_{✳ ❙✐♥❝❡ ❘❡✐ss♥❡r✲▼✐♥❞❧✐♥ ❦✐♥❡♠❛t✐❝s ✐s ❛ss✉♠❡❞✱} t❤❡ ❝r♦ss✲s❡❝t✐♦♥ r♦t❛t✐♦♥ ✐s ❝♦♥s✐❞❡r❡❞ ❛♥ ✐♥❞❡♣❡♥❞❡♥t q✉❛♥t✐t②✳ ❚❤❡ ❡✛❡❝t✐✈❡ r♦t❛t✐♦♥θ

❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❊✉❧❡r ♣❛r❛♠❡t❡rs ❣r♦✉♣❡❞ ✐♥ t❤❡ r♦t❛t✐♦♥ ✈❡❝t♦r t❤r♦✉❣❤ t❤❡ ❊✉❧❡r✲ ❘♦❞r✐❣✉❡s r❡❧❛t✐♦♥✿

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✼

✇❤❡r❡

Θ =❙❦❡✇(θ), ✭✷✳✶✸❛✮

θ =kθk, ✭✷✳✶✸❜✮

h1(θ) = s✐♥(θ)

θ , ✭✷✳✶✸❝✮

h2(θ) =

1 2

s✐♥ θ 2

θ 2

!2

. ✭✷✳✶✸❞✮

❚❤❡ ❝✉rr❡♥t ♣♦s✐t✐♦♥ ♦❢ ❛ ♠❛t❡r✐❛❧ s❤❡❧❧ ♣♦✐♥t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❜②

x =z +a =zo+u +Qeao. ✭✷✳✶✹✮ ❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ u ❛❧♦♥❣ ✇✐t❤ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ θ ✐♥ ❛ ❣❧♦❜❛❧ ❈❛rt❡s✐❛♥ s②st❡♠

❞❡✜♥❡ t❤❡ s✐① ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ♦❢ t❤❡ t❤❡♦r②✳

❚❤❡ ❞❡❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ✢❛t r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ t♦ t❤❡ ❝✉rr❡♥t ❞❡❢♦r♠❡❞ ❝♦♥✜❣✲ ✉r❛t✐♦♥ ✐s ❝❛❧❧❡❞ t❤❡ t♦t❛❧ ❞❡❢♦r♠❛t✐♦♥✱ ❛♥❞ ✐ts ❣r❛❞✐❡♥t ✐s ❣✐✈❡♥ ❜②

F = ∂x

∂ξ =Q + (ηα+κα×a)⊗e

r

α ✭✷✳✶✺✮

✇❤❡r❡✱ ❛s ✐t ✇❛s ❞♦♥❡ ❢♦r t❤❡ ✐♥✐t✐❛❧ ❞❡❢♦r♠❛t✐♦♥✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡

η_α=z,α−eα =zo,α+u,α−Qerα ❛♥❞

κα=❛①✐❛❧ Q,αQT

=κe_α+Qeκo_α, ✭✷✳✶✻✮ ❛s t❤❡ ❝r♦ss✲s❡❝t✐♦♥❛❧ str❛✐♥ ✈❡❝t♦rs✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡✐r ❜❛❝❦✲r♦t❛t❡❞ ❝♦✉♥t❡r♣❛rts ❛r❡ ♠♦r❡ s✉✐t❛❜❧❡ ❢♦r ❛♥ ♦❜❥❡❝t✐✈❡ t❤❡♦r②✳ ❆❢t❡r s♦♠❡ ❛❧❣❡❜r❛✱ ✇❡ ❛rr✐✈❡ ❛t

ηr_α =QTηα =QTz,α−erα ❛♥❞

κr_α =QTκα=κerα +κ or α,

✭✷✳✶✼✮

✇❤❡r❡

κerα =Γ T

θ,α, ✇✐t❤

Γ =I +h2(θ)Θ+h3(θ)Θ 2

❛♥❞ h3(θ) = 1−h1(θ)

θ2 .

✭✷✳✶✽✮

❋✐♥❛❧❧② t❤❡ ❜❛❝❦✲r♦t❛t❡❞ str❛✐♥ ✈❡❝t♦rs ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s

εrα =

"

ηr α

κr α

#

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✽

❛♥❞ t❤❡ t♦t❛❧ ❜❛❝❦✲r♦t❛t❡❞ str❛✐♥s ❛s

γr_α =ηr_α+κr_α×ar, ✭✷✳✷✵✮ s♦ t❤❡ t♦t❛❧ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❜②

F =Q(I +γr_α⊗er_α). ✭✷✳✷✶✮ ■ts ❜❛❝❦✲r♦t❛t❡❞ ❝♦✉♥t❡r♣❛rt✱ ♥♦t ❛✛❡❝t❡❞ ❜② r✐❣✐❞ ❜♦❞② ♠♦t✐♦♥s ✐s

Fr=QTF =I +γr_α⊗er_α ✭✷✳✷✷✮

❚❤✐s t♦t❛❧ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ✐♥✐t✐❛❧ ❛♥❞ ❡✛❡❝t✐✈❡ ♦♥❡s

F =FeFo,Fe=F Fo−1 _=Q(fer

i ⊗e o

i). ✭✷✳✷✸✮

❚❤❡ ❡✛❡❝t✐✈❡ ❜❛❝❦✲r♦t❛t❡❞ str❛✐♥s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s

γerα =f er

α −erα =Jo− 1

er_α·gor_β er_β+γrβ

−er_α, ✭✷✳✷✹✮ ✇❤❡r❡gor

β ❛r❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞Jo ✐s ✐ts ❏❛❝♦❜✐❛♥✱ ❛s ❢♦❧❧♦✇s Jo _=❞❡tFo

=❞❡tFo =er3·f

or 1 ×f

or

2 , ✭✷✳✷✺❛✮

gor1 =f or 2 ×e

r

3,gor2 =e r 3×f

or 1 ,g

or 3 =f

or 1 ×f

or

2 . ✭✷✳✷✺❜✮

✷✳✸ ❙t❛t✐❝s

❚❤❡ s❤❡❧❧ s♣❡❝✐✜❝ ✐♥t❡r♥❛❧ ♣♦✇❡r ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❛t t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❜②

P : ˙F =Jo_{P : ˙F}e_=τr α·η˙

r α+ (a

r_×τr α)·κ˙

r

α, ✭✷✳✷✻✮

✇❤❡r❡ P =τα⊗erα ✐s t❤❡ ✜rst P✐♦❧❧❛✲❑✐r❝❤❤♦✛ str❡ss t❡♥s♦r✱ τrα = Q oT

τα ❛r❡ ✐ts ❜❛❝❦✲ r♦t❛t❡❞ ❝♦❧✉♠♥ ✈❡❝t♦rs ❛♥❞ _F˙ _{✐s t❤❡ ✈❡❧♦❝✐t② ❣r❛❞✐❡♥t✳ ■♥t❡❣r❛t✐♦♥ ❛❝r♦ss t❤❡ t❤✐❝❦♥❡ss}

②✐❡❧❞s t❤❡ ✐♥t❡r♥❛❧ ♣♦✇❡r ♣❡r ✉♥✐t ❛r❡❛ ♦❢ t❤❡ ♠✐❞s✉r❢❛❝❡ ✐♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥

Z

H

P : ˙FdH =nr_α·η˙r_α+mr_α·κ˙r_α, ✭✷✳✷✼✮

✇❤❡r❡H = [−h_/2,h_{/2] ✐s t❤❡ t❤✐❝❦♥❡ss ❞♦♠❛✐♥ ❛♥❞ t❤❡ ✈❡❝t♦rs}

nr_α = Z

H

(τrα)dH ❛♥❞ m

r α =

Z

H

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✾

❛r❡ t❤❡ ❜❛❝❦✲r♦t❛t❡❞ ❢♦r❝❡ ❛♥❞ ♠♦♠❡♥t r❡s✉❧t❛♥ts✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤♦s❡ ❝❛♥ ❜❡ ❣❛t❤❡r❡❞ ✐♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❜❛❝❦✲r♦t❛t❡❞ str❡ss ✈❡❝t♦rs

σr_α = " nr α mr α # ✭✷✳✷✾✮

s♦ t❤❛t t❤❡ ✐♥t❡r♥❛❧ t❤❡ ✐♥t❡r♥❛❧ ♣♦✇❡r ♦❢ t❤❡ s❤❡❧❧ ✐s

Pint ₌

Z

Vr

P : ˙FdVr ₌

Z

Ωr

(σr_α·ε˙r_α)dΩr_{. ✭✷✳✸✵✮} ❚❤❡ ❡①t❡r♥❛❧ ♣♦✇❡r ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ ❜②

Pext₌

Z

Ωr

¯

qΩr ·d˙dΩr₊

Z

Γr t

¯ qΓtr ·d˙

dΓt+

Z

Γr u

qΓur ·_d˙

dΓr

u ✭✷✳✸✶✮

✇❤❡r❡

¯ qΩr =

" ¯ nΩr ΓTm¯Ωr

#

✭✷✳✸✷✮ ❛r❡ t❤❡ ❢♦r❝❡ ❛♥❞ ♠♦♠❡♥t r❡s✉❧t❛♥ts ♦❢ t❤❡ ❧♦❛❞s ❛♣♣❧✐❡❞ ✐♥ t❤❡ ❞♦♠❛✐♥✱

¯ qΓtr =

" ¯ nΓr

t

ΓTm¯Γr t #

✭✷✳✸✸✮

✐♥ t❤❡ ♥❛t✉r❛❧ ❜♦✉♥❞❛r② ❛♥❞

qΓur = "

nΓr u

ΓTmΓur #

✭✷✳✸✹✮ ❛r❡ t❤❡ ✭✉♥❦♥♦✇♥✮ r❡❛❝t✐♦♥s ✐♥ t❤❡ ❡ss❡♥t✐❛❧ ❜♦✉♥❞❛r②Γr

u✳ ❚❤♦s❡ ✭tr✉❡✮ str❡ss r❡s✉❧t❛♥ts ❛r❡ ❞❡✜♥❡❞ ❛s

¯

nΩr = ¯ttr+ ¯tbr + Z

h

¯ brdh

¯

mΩr =aׯtbr+ Z

h

a×b¯rdh

✭✷✳✸✺❛✮

♦♥ t❤❡ ❞♦♠❛✐♥✱

¯ nΓtr =

Z

h

¯ tlrdh

¯ mΓtr =

Z

h

aׯtlrdh

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✶✵

♦♥ t❤❡ ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r②✱ ❛♥❞

nΓur = Z

h

rdh

mΓur = Z

h

a ×rdh

✭✷✳✸✺❝✮

♦♥ t❤❡ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r②✳

■♥ ✭✷✳✸✺✮✱ ¯tt✱ ¯tb✱ ¯tl ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❡①t❡r♥❛❧ s✉r❢❛❝❡ tr❛❝t✐♦♥s ♦♥ t❤❡ t♦♣✱ t❤❡

❜♦tt♦♠ ❛♥❞ t❤❡ ❧❛t❡r❛❧ ♦❢ t❤❡ s❤❡❧❧✱ ♣❡r ✉♥✐t r❡❢❡r❡♥❝❡ ❛r❡❛✱ _b¯ _{✐s t❤❡ ❜♦❞② ❢♦r❝❡ ✈❡❝t♦r}

♣❡r ✉♥✐t r❡❢❡r❡♥❝❡ ✈♦❧✉♠❡ ❛♥❞ r ✐s t❤❡ r❡❛❝t✐♦♥ ♦♥ t❤❡ ❧❛t❡r❛❧ s✉r❢❛❝❡ ❛t t❤❡ ❦✐♥❡♠❛t✐❝

❜♦✉♥❞❛r②✳

✷✳✹ ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❡ ❝♦♥s✐st❡♥t ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ t♦ ❛ ✜rst ♦r❞❡r ✐♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ r♦t❛t✐♦♥s ✇❛s ❝❛rr✐❡❞ ♦✉t ✐♥ ❬✶✷❪✳ ❚❤❡ ♦❜t❛✐♥❡❞ ♠♦❞❡❧ ✐s s✐♠✐❧❛r t♦ ❛ t❤✐❝❦ ♣❧❛t❡ t❤❡♦r②✱ ❡♥❝♦♠♣❛ss✐♥❣ s❤❡❧❧ ❝✉r✈❛t✉r❡ ❛♥❞ ♠❡♠❜r❛♥❡ ❞❡❢♦r♠❛t✐♦♥✳

❚♦ t❤❡ ✜rst ♦r❞❡r ✐♥ θ✱ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t δ ❞✉r✐♥❣ t❤❡ ❡✛❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❛

❣✐✈❡♥ ♣♦✐♥t ✐♥ t❤❡ s❤❡❧❧ ✐s

δ =u +θ×ao. ✭✷✳✸✻✮

❇❛❝❦✲r♦t❛t❡❞ ✭♦❜❥❡❝t✐✈❡✮ ❣❡♥❡r❛❧✐③❡❞ ♠❡♠❜r❛♥❡ ❛♥❞ ❝✉r✈❛t✉r❡ str❛✐♥ q✉❛♥t✐t✐❡s ❬✽✱ ♣✳ ✷✵✹✸❪ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❊q✳ ✷✳✶✼ t♦ t❤❡ ✜rst ♦r❞❡r ✐♥u ❛♥❞ θ

ηr_α =QoT z_,αo +Zo_,αθ+u,α

−er_α ❛♥❞ ✭✷✳✸✼❛✮

κrα =Q oT

κo,α+θ,α

. ✭✷✳✸✼❜✮

✇❤❡r❡

Zo_,α =Skew zo_,α. ✭✷✳✸✽✮

■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❡ ♥❡❛t ❛❞❞✐t✐✈❡ ♣r♦♣❡rt②

ηr_α=ηor_α +ηer✱ ✇❤❡r❡ ✭✷✳✸✾❛✮ ηor_α =QoTzo_,α−er_α ❛♥❞ ✭✷✳✸✾❜✮

ηerα =Q oT

Zo_,αθ+QoTu,α ✭✷✳✸✾❝✮

❛♥❞

κrα =κorα +κer✱ ✇❤❡r❡ ✭✷✳✹✵❛✮

κor_α =QoTκo_α = eo2,α·e o 3

er1 + e o 3,α·e

o 1

er2+ e o 1,α·e

o 2

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✶✶

κer_α =QoTθ,α, ✭✷✳✹✵❝✮

♦❢ t❤❡ ❣❡♥❡r❛❧✐③❡❞ str❛✐♥s✳ ■t✬s ❛❧s♦ ❝♦♠♣❡❧❧✐♥❣ t♦ s♣❧✐t t❤❡ ❣❡♥❡r❛❧✐③❡❞ str❛✐♥ ✈❡❝t♦rs εr α ❣✐✈❡♥ ✐♥ ✭✷✳✶✾✮ ✐♥t♦ ✐♥✐t✐❛❧ ❛♥❞ ❡✛❡❝t✐✈❡ str❛✐♥s

εr_α=εor_α +εer_α. ✭✷✳✹✶✮

❚❤❡ ✐♥✐t✐❛❧ str❛✐♥s εor

α ❛r❡ ❣✐✈❡♥ ❜② ✭✷✳✸✾❜✮ ❛♥❞ ✭✷✳✹✵❜✮ ❛♥❞ εerα ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ ♠❛tr✐① ❢♦r ❛s

εer_α =∆αd, ✭✷✳✹✷❛✮

∆α =

"

QoT 0

0 QoT

# " I_∂∂_ξ

α Z

o ,α

0 I_∂∂_ξ α

#

, ✭✷✳✹✷❜✮

✇❤❡r❡ t❤❡ s❤❡❧❧✬s ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❛r❡ ❝♦❧❧❡❝t❡❞ ✐♥

d =

" u

θ

#

. ✭✷✳✹✸✮

✷✳✺ ▼❛t❡r✐❛❧

❆ ❣❡♥❡r❛❧✐③❡❞ ❍♦♦❦❡ ♠❛t❡r✐❛❧ ✐s ✉s❡❞✳ ❲✐t❤ t❤❡ ❧✐tt❧❡ ❛❜✉s❡ ♦❢ ♥♦♠❡♥❝❧❛t✉r❡ ♠❛❞❡ ✐♥ ♠♦st ♦❢ t❤❡ ❝♦♥s✉❧t❡❞ ❧✐t❡r❛t✉r❡✱ t❤❡ ♠♦❞❡❧ ♣r❡s❡♥t❡❞ ❤❡r❡ ✐s s❛✐❞ t♦ ❜❡ s✉❜❥❡❝t t♦ ❛ ♣❧❛♥❡ str❡ss ❛ss✉♠♣t✐♦♥✳ ❆❝t✉❛❧❧②✱ t❤❡ ♠♦❞❡❧ ❛❝❝♦✉♥ts ❢♦r s❤❡❛r str❡ss❡s ✐♥ t❤❡ t❤✐❝❦♥❡ss ❞✐r❡❝t✐♦♥✱ ✇❤❛t ❜② ✐ts❡❧❢ ✐s ❛ ✈✐♦❧❛t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ str❡ss ❝♦♥❞✐t✐♦♥✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♠❛❞❡ ❤❡r❡✐♥ ✐s t❤❛t t❤❡r❡ ✐s ♥♦ ♥♦r♠❛❧ str❡ss ❛❝t✐♥❣ ♦♥ t❤❡ ♠✐❞s✉r❢❛❝❡ ♣❧❛♥❡ ✭er

3·P r_er

3 = 0✮✱ ❛s ✐t ❤❛s ❜❡❡♥ ❞♦♥❡ ✐♥ ❬✶✸❪ ❢♦r t❤❡ ♣♦❧②❝♦♥✈❡① ♥❡♦✲❍♦♦❦❡❛♥ ❙✐♠♦✲❈✐❛r❧❡t ♠❛t❡r✐❛❧ ✭s❡❡ ❬✺✺❪✱ ♣❛❣❡ ✷✺✽✮✳ ❚❤✉s✱ ✐t ✐s st❛t❡❞ t❤❛t

τer_α =Cer_αβγer_β , ✇❤❡r❡ ✭✷✳✹✹❛✮

Cer_αβ = E

2 (1−ν)e

r α⊗e

r β+

E

2 (1 +ν)ǫαβE

r 3+

E

2 (1 +ν)δαβI, ✭✷✳✹✹❜✮

✇❤✐❝❤ ✐s t❤❡ ✐s♦tr♦♣✐❝ ♣❧❛♥❡ str❡ss ❣❡♥❡r❛❧✐③❡❞ ❍♦♦❦❡✬s ❧❛✇ ✐♥ ♦✉r ♥♦t❛t✐♦♥✳ ■♥ ✭✷✳✹✹❜✮✱ t❤❡ s❦❡✇✲s②♠♠❡tr✐❝ t❡♥s♦rEr3 =Skew(er3)✱ t❤❡ ❑r♦♥❡❝❦❡r s②♠❜♦❧ δαβ ❛♥❞ t❤❡ ♣❡r♠✉t❛t✐♦♥ s②♠❜♦❧ ǫαβ = er

α ·erβ × er3 ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞✳ ❚❤❡ ❧✐♥❡❛r ❡❧❛st✐❝ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❜❛❝❦✲r♦t❛t❡❞ str❡ss❡s σr

α ❛♥❞ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❡✛❡❝t✐✈❡ str❛✐♥s εerα ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❞✐s♣❧❛②❡❞ ❜❡❧♦✇✳

σr_α =Dαβεer

β =Dαβ εrβ−ε or β

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✶✷

e r 2 er₃

er₁

eo₁ e

o 2 e o 3 x1 x2 (a) e r 2 er₃

er₁

e o 2 e o 1 e o 3 x1 x2 r (b) e r 2 er₃

er₁

eo₂ eo₁

e o 3 x1 x2 (c)

❋✐❣✉r❡ ✷✳✷✿ ■♥✐t✐❛❧ ❞❡❢♦r♠❛t✐♦♥s ✭❛✮ ✢❛t s❤❡❧❧✱ ✭❜✮ ❝②❧✐♥❞r✐❝❛❧ s❤❡❧❧ ❛♥❞ ✭❝✮ s♣❤❡r✐❝❛❧ s❤❡❧❧

✇❤❡r❡

Dαβ =

"

Dηη_αβ Dηκ_αβ Dκη_αβ Dκκ_αβ #

, ✭✷✳✹✻✮

✇✐t❤

Dηη_αβ = Z

H

1

Jo (g or α ·e

r δ)C

er δγ e

r γ·g

or β

dξ3,

Dηκ_αβ =−

Z

H

1

Jo (g or

α ·erδ)C er

δγ erγ·gorβ

Ardξ3

=DκηT_αβ ❛♥❞

Dκκ_αβ = Z

H

1

JoA r

(gor_α ·er_δ)Cer_δγ er_γ·gor_β

Ardξ3.

✭✷✳✹✼✮

■♥ ✭✷✳✹✼✮ t❤❡ s❦❡✇ s②♠♠❡tr✐❝ t❡♥s♦r Ar3 = Skew(ar) ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❛♥❞ gorα ❛r❡ t❤❡ ❝♦❧✉♠♥ ✈❡❝t♦rs ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t For_i ⊗er_i ❣✐✈❡♥ ✐♥

❊q✳ ✷✳✷✺❜✳

✷✳✻ ■♥✐t✐❛❧ ❉❡❢♦r♠❛t✐♦♥

❚❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ t❤r♦✉❣❤ t❤❡ t❤✐❝❦♥❡ss ✐♥t❡❣r❛❧s ✐♥ t❤❡ ♠❛t❡r✐❛❧ ♠❛tr✐① ❝❛♥ ❜❡ ♥✉♠❡r✐❝❛❧❧② ♣❡r❢♦r♠❡❞ ❢♦r ❛♥② ✐♥✐t✐❛❧ ♠❛♣♣✐♥❣✱ ✐♥❝❧✉❞✐♥❣ ♣❛r❛♠❡tr✐❝ ❛♣♣r♦①✐♠❛t✐♦♥s s✉❝❤ ❛s ◆❯❘❇❙ ♦r ❚✲s♣❧✐♥❡s✳ ❚❤✐s ❝❛♥ ❜❡ ❛✈♦✐❞❡❞ ❢♦r s♦♠❡ ✐♥✐t✐❛❧ s❤❡❧❧ s❤❛♣❡s t❤r♦✉❣❤ ❛♥ ❛♥❛❧②t✐❝❛❧ ❡✈❛❧✉❛t✐♦♥✳ ❙♦♠❡ ♦❢ t❤❡ ♦❜t❛✐♥❡❞ ❡①♣r❡ss✐♦♥s ❝❛♥ ❜❡ ❢✉rt❤❡r s✐♠♣❧✐✜❡❞ ✐❢ ❛ t❤✐♥♥❡ss ❛ss✉♠♣t✐♦♥ ✐s ✐♥tr♦❞✉❝❡❞✱ ✇❤✐❝❤ ✐s ❤❡r❡ ❞❡✜♥❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ h

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✶✸

❛♥❞ ♥✉♠❡r✐❝❛❧ ♠❛♣♣✐♥❣s ❛r❡

Dηη_αβ =hJo−1(_gor

α ·e r δ) e

r γ·g

or β

Cer_δγ,

Dηκ_αβ =DκηT_αβ =0 ❛♥❞ Dκκ_αβ =−h

3

12J

o−1_(gor

α ·e r δ) e

r γ·g

or β

Er3C

er δγE

r 3,

✭✷✳✹✽✮

✇✐t❤ Jo _{❛♥❞ g}or

α ❡✈❛❧✉❛t❡❞ ♦♥ t❤❡ s❤❡❧❧ ♠✐❞s✉r❢❛❝❡✳

✷✳✼ ❲❡❛❦ ❋♦r♠ ♦❢ ❊q✉✐❧✐❜r✐✉♠

❊q✉✐❧✐❜r✐✉♠ ❝❛♥ ❜❡ ❛ss❡rt❡❞ ✐♥ ❛ ✇❡❛❦ ❢♦r♠ ❛s st❛t✐♥❣ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ✐♥t❡r♥❛❧ ❛♥❞ ❡①t❡r♥❛❧ ✇♦r❦s ✐s ③❡r♦✳ ❲❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ✐♥t❡r♥❛❧ ✈✐rt✉❛❧ ✇♦r❦ ❢r♦♠ ✭✷✳✸✵✮ ❛s

δWint ₌

Z

Γr u

σrα·δεrα dΩr. ✭✷✳✹✾✮ ❋r♦♠ t❤❡ ❡①t❡r♥❛❧ ♣♦✇❡r ✭✷✳✸✶✮✱ ✇❡ s❡♣❛r❛t❡ t❤❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤❡ ❡①t❡r♥❛❧ ❧♦❛❞s ✐♥t♦ t❤❡ ❡①t❡r♥❛❧ ✈✐rt✉❛❧ ✇♦r❦

δWext₌

Z

Ωr ¯

qΩr ·δd dΩr₊

Z

Γr t

¯

qΓtr ·δd dΓr

t ✭✷✳✺✵✮

❛♥❞ t❤❡ ❡①t❡r♥❛❧ ✈✐rt✉❛❧ ✇♦r❦ ♦❢ t❤❡ r❡❛❝t✐♦♥ ❢♦r❝❡s

δWrea₌

Z

Γr u

qΓur ·δ_d dΓr

u. ✭✷✳✺✶✮

❚❤❛t s❡♣❛r❛t✐♦♥ ✇✐❧❧ ❝♦♠❡ ✐♥ ❤❛♥❞② ✇❤❡♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ✐♠♣♦s✐t✐♦♥ ♦❢ ❡ss❡♥t✐❛❧ ❜♦✉♥❞✲ ❛r② ❝♦♥❞✐t✐♦♥s✳ ❊q✉✐❧✐❜r✐✉♠ ❝❛♥ t❤❛♥ ❜❡ st❛t❡❞ ✐♥ ❛ ✇❡❛❦ ❢♦r♠ ❛s

δWreg _=δWint_−δWext_−δWrea _{= 0. ✭✷✳✺✷✮} ❖r ✐♥ ❛ ♠♦r❡ ❡①♣r❡ss✐✈❡ ✇❛② ❛s ✏✜♥❞ d ∈H 1

(Ω)s✉❝❤ t❤❛t d = ¯d ✐♥ Γd ❛♥❞

δWreg ₌

Z

Ωr

σr_α·δεr_αdΩr₋

Z

Ωr ¯

qΩr ·δddΩr

−

Z

Γr t

¯

qΓtr ·δddΓr

t −

Z

Γr u

qΓur ·δddΓr

u = 0

✭✷✳✺✸✮

❢♦r ❛❧❧δd ∈H 1

(Ω)✑✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ✉s✐♥❣ ❛ ♥♦♥✲✐♥t❡r♣♦❧❛t♦r② ❛♣♣r♦①✐♠❛t✐♦♥

δd ❝❛♥♥♦t ❜❡ ♠❛❞❡ t♦ ✈❛♥✐s❤ ♦♥ t❤❡ ❡ss❡♥t✐❛❧ ❜♦✉♥❞❛r②✱ t❤✉s t❤❡ ❧❛st t❡r♠ ✭✐♥ Γr

u✮ ✐s ♥♦t ❝❛♥❝❡❧❡❞ ❧✐❦❡ ✐♥ ❛ ❋✐♥✐t❡ ❊❧❡♠❡♥t ❢r❛♠❡✇♦r❦✳

❚❤✐s ✐♠♣♦s✐t✐♦♥ ✐s ❧✐♥❡❛r ♦♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❞✐s♣❧❛❝❡♠❡♥ts ❛s t❤❡ str❡ss❡sσr

α❛r❡ ❧✐♥❡❛r❧② ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ str❛✐♥sεr

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✶✹ o 4 F r 4

W

W

❋✐❣✉r❡ ✷✳✸✿ ▼✉❧t✐✲r❡❣✐♦♥ ♣r♦❜❧❡♠✳

✉s✐♥❣ ✭✷✳✹✷✮✳

✷✳✽ ▼✉❧t✐✲r❡❣✐♦♥ s❤❡❧❧s

❆ ♠✉❧t✐✲r❡❣✐♦♥ ♣r♦❜❧❡♠ ❛r✐s❡s ✇❤❡♥ t❤❡ ♣r♦❜❧❡♠ ❞♦♠❛✐♥ ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ s✐♥❣❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ r❡❣✐♦♥ ❢♦r ❧❛❝❦✐♥❣ ♠❛t❡r✐❛❧ ❝♦♥t✐♥✉✐t②✱ ✉s✉❛❧❧② ❞✉❡ t♦ ❝❤❛♥❣❡ ✐♥ ♠❛✲ t❡r✐❛❧ ♣❛r❛♠❡t❡rs s✉❝❤ ❛s ❨♦✉♥❣ ♠♦❞✉❧✉s ♦r t❤✐❝❦♥❡ss❀ ♦r ✇❤❡♥ ❞✐✛❡r❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt✐❡s ❛r❡ r❡q✉✐r❡❞✱ ❛s ❝♦✉♣❧✐♥❣ ❞✐✛❡r❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ✭❡✳❣✳ ❊❋● ❝♦✉♣❧❡❞ t♦ ❋❊s✱ ♦r ■s♦❣❡♦♠❡tr✐❝ ◆❯❘❇❙✮ ♦r ❞✐✛❡r❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt✐❡s ❧✐❦❡ ♣♦❧②♥♦♠✐❛❧ ❜❛s❡ ♦r s✉❞❞❡♥ ♣❛rt✐❝❧❡ ❞❡♥s✐t②✳ ■♥ ♠❛♣♣❡❞ s❤❡❧❧ ❛♥❛❧②s✐s t❤❡ ✐♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❝♦♠❡s ❢r♦♠ ❛ ❑✐r❝❤❤♦✛✲▲♦✈❡ ♠❛♣♣✐♥❣✱ ✇❤✐❝❤ r❡q✉✐r❡ t❤❡ s❤❡❧❧ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s ✇✐t❤ ❛❧s♦ ❝♦♥✲ t✐♥✉♦✉s ✜rst ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ♣r❡s❡♥❝❡ ♦❢ ❢♦❧❞s ♦r t❤❡ ❝♦♥♥❡❝t✐♦♥ ♦❢ ♠✉❧t✐♣❧❡ s❤❡❧❧s ❛❧♦♥❣ ❛ ❧✐♥❡ ♣♦s❡s ❛ ♠❛t❡r✐❛❧ ❞✐s❝♦♥t✐♥✉✐t②✱ ❛♥❞ t❤❡ ♠♦❞❡❧ s❤♦✉❧❞ ❡♥❝♦♠♣❛ss t❤❡s❡ ❥✉♠♣s✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s❤♦✉❧❞ t❤❡♥ ❜❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ r♦t❛t✐♦♥s✱ ❜✉t ❞✐s❝♦♥t✐♥✲ ✉♦✉s ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ str❛✐♥s✳ ❚❤✐s ✐s s❛✐❞ t♦ ❜❡ ❛ ✇❡❛❦ ❞✐s❝♦♥t✐♥✉✐t②✱ ❛s ♦♣♣♦s❡❞ t♦ ❛ str♦♥❣ ❞✐s❝♦♥t✐♥✉✐t② ♦♥ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✜❡❧❞s✳

▲❡t t❤❡ ✐♥✐t✐❛❧ ❞♦♠❛✐♥Ωo_{✐♥ ❋✐❣✳ ✷✳✸ ❜❡ s♣❧✐t ✐♥t♦nreg ♠✉❧t✐♣❧❡ r❡❣✐♦♥sΩ}o_{n✳ ❊❛❝❤ r❡❣✐♦♥} ✇✐❧❧ ❤❛✈❡ ❜♦✉♥❞❛r✐❡sΓo

un ✇❤❡r❡ ✐t ♠❡❡ts t❤❡ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r②✱Γtno ❛❧♦♥❣ ✐ts ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ t❤❡ ◆❡✇♠❛♥ ❜♦✉♥❞❛r② ❛♥❞ ✐♥t❡r❢❛❝❡ ❜♦✉♥❞❛r② Γo

in ✇❤❡r❡ ✐t ❝♦♥t❛❝ts ♦t❤❡r r❡❣✐♦♥s ✭✐✳❡✳ Ωo

1✮✳ ❘❡❣✐♦♥s ♠❛② ❤❛✈❡ ♠♦r❡ t❤❛♥ ♦♥❡ ✐♥t❡r❢❛❝❡ ❜♦✉♥❞❛r②✱ ♦r ✐♥t❡r❛❝t ✇✐t❤ ♦t❤❡r r❡❣✐♦♥ r❛t❤❡r t❤❛♥Ωo_{1✱ ❜✉t ✐♥ ♦r❞❡r t♦ ❦❡❡♣ ❛ s✐♠♣❧❡ ♥♦t❛t✐♦♥✱ t❤♦s❡ ❡❛s✐❧② ❞❡r✐✈❛❜❧❡ ❝❛s❡s} ✇✐❧❧ ❜❡ ♦♠✐tt❡❞✳

✷✳ ▲✐♥❡❛r s❤❡❧❧ t❤❡♦r② ✶✺

r❛t✐♦♥Ω_{n✳ ❖✉r s❤❡❧❧ ❢♦r♠✉❧❛t✐♦♥ ♣❡r❢♦r♠s ✐♥t❡❣r❛t✐♦♥s ♦✈❡r t❤❡s❡ r❡❢❡r❡♥❝❡ ❞♦♠❛✐♥s ❛♥❞}r t❤❡ r❡s♣❡❝t✐✈❡ ❜♦✉♥❞❛r✐❡s ✭Γ_un✱r _Γr

tn ❛♥❞ Γin✮✳ ❆♥ ✐♥t❡r❢❛❝❡ r❡❢❡rs t♦ t❤❡ ❧✐♥❡ ✇❤❡r❡ ❞✐❢✲r ❢❡r❡♥t s❤❡❧❧s✬ ♠✐❞s✉r❢❛❝❡s ✐♥t❡rs❡❝t ❛♥❞ t❤❡ ✐♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥ ✐s t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥ts ❛♥❞ r♦t❛t✐♦♥s ❝♦♠✐♥❣ ❢r♦♠ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ✐♥ ❜♦t❤ r❡❣✐♦♥s

d|Ω₁ =d|Ω₂ ✭✷✳✺✹✮

❛♥❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❜❡t✇❡❡♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ str❡ss❡s ♥♦r♠❛❧ t♦ t❤❡ ✐♥t❡r❢❛❝❡ ✐♥ ❜♦t❤ r❡❣✐♦♥s✳

❈❤❛♣t❡r ✸

▼❡s❤❧❡ss ❆♣♣r♦①✐♠❛t✐♦♥

❆♠♦♥❣ t❤❡ ♠❛♥② ♠❡s❤❧❡ss ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡s ❞❡s❝r✐❜❡❞ ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✸✾✱ ✹✵❪ ✱ t❤❡ ▼♦✈✐♥❣ ▲❡❛st ❙q✉❛r❡s ✭▼▲❙✮ ✇✐❧❧ ❜❡ ✉s❡❞ ✐♥ t❤❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s✳ ❙t✐❧❧✱ t❤❡ s❤❡❧❧ ♠♦❞❡❧ ❛♥❞ ❝♦♥s❡q✉❡♥t ❞✐s❝r❡t❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ♠❛② ❜❡ ✉s❡❞ ✇✐t❤ ❛♥② ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡✱ ❡s♣❡❝✐❛❧❧② ♥♦♥✲✐♥t❡r♣♦❧❛t♦r② ♦♥❡s✳

❚❤❡ s♦ ❝❛❧❧❡❞ ❉✐✛✉s❡ ❆♣♣r♦①✐♠❛t✐♦♥ ✇❛s ✉s❡❞ ❜② ◆❛②r♦❧❡s ❛t ❛❧✳ ❬✹✺❪ ❢♦r ♣❧❛♥❡ ❡❧❛st✐❝✐t② ♣r♦❜❧❡♠s✳ ❚❤❡ s❡♠✐♥❛❧ ♣❛♣❡r ❬✻❪ ✐❞❡♥t✐✜❡s t❤❡ ❢♦r♠✉❧❛t✐♦♥ ❛s t❤❡ s❛♠❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✉s❡❞ ❢♦r s❝❛tt❡r❡❞ ❞❛t❛ ✜tt✐♥❣ ❜② ▲❛♥❝❛st❡r ❛♥❞ ❙❛❧❦❛✉s❦❛s✱ ▼❝▲❛✐♥✱ ●♦r❞♦♥ ❛♥❞ ❲✐①♦♠✱ ❇❛r♥❤✐❧❧ ❛♠♦♥❣ ♦t❤❡rs✱ ❝❛❧❧❡❞ ▼♦✈✐♥❣ ▲❡❛st ❙q✉❛r❡s✳

▼❛② u(x) ❜❡ ❛ ✜❡❧❞ ✈❛r✐❛❜❧❡ t♦ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ♦✈❡r Ω✳ ■♥ t❤❡ ▼▲❙ ❢r❛♠❡✇♦r❦✱ ✐ts ❛♣♣r♦①✐♠❛t✐♦♥ uh _{❛r♦✉♥❞ ❛ ♣♦✐♥t x ∈ Ω ✐s ❡①♣r❡ss❡❞ ❜② ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ s❡t ♦❢} ❢✉♥❝t✐♦♥s✱ t❤❡ ❜❛s✐sp(x)✿

uh_{(x) = p(x)·a(x), ✭✸✳✶✮} ✇✐t❤ ❝♦❡✣❝✐❡♥tsa t❤❛t ✈❛r② ❛❝r♦ss t❤❡ ❞♦♠❛✐♥✳ ■❢ ❛ s❡t ♦❢ ♣❛rt✐❝❧❡sx¯i✱ ♣♦s✐t✐♦♥s ♦✈❡r t❤❡

❞♦♠❛✐♥ ♦r ❝❧♦s❡ t♦ ✐t✱ ✐s ❛ss♦❝✐❛t❡❞ t♦ ❛ s❡t ♦❢ ♥♦❞❛❧ ✈❛❧✉❡s u¯i✱ t❤❡ ❝♦❡✣❝✐❡♥ts a(x) ❝❛♥

❜❡ ♦❜t❛✐♥❡❞ t❤r♦✉❣❤ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ✇❡✐❣❤t❡❞ ❡rr♦r

J(a) =

n

X

i=1

w(r) ¯ui−uh_(¯x i)

2

, ✭✸✳✷✮

r = kx −x¯k

rmax ✭✸✳✸✮

✇✐t❤ r❡s♣❡❝t t♦ a✱ ✇❤❡r❡ w ✐s ❛ ✇❡✐❣❤t✐♥❣ ❢✉♥❝t✐♦♥ t❤❛t ❡♥❢♦r❝❡s t❤❡ ❧♦❝❛❧ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ■t ✐s ✉s✉❛❧❧② ❛ ❜❡❧❧✲s❤❛♣❡❞ ❢✉♥❝t✐♦♥ ✇✐t❤ s✉♣♣♦rt r❛❞✐✉s ✶✱ t❤❛t ♠♦♥♦t♦♥✐❝❛❧❧② ❞❡❝r❡❛s❡s ❢r♦♠ t❤❡ ✉♥✐t ❛tr= 0 t♦ ③❡r♦ ✇❤❡♥r ≥1✳ ❚❤❡ r❡q✉✐r❡♠❡♥ts ❢♦r

t❤✐s ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ❛❧♦♥❣ ✇✐t❤ ❡①❛♠♣❧❡s ✐♥ ❬✺✽❪✳ ❋♦r t❤❡ ♣r❡s❡♥t ❝♦♥t❡①t✱ ✐t s✉✣❝❡s