4.2 Differential Equations in Polar Coordinates
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1 Section Diffeential qations in Pola Coodinates Hee the two-dimensional Catesian elations of Chapte ae e-cast in pola coodinates. 4.. qilibim eqations in Pola Coodinates One wa of epesg the eqations of eqilibim in pola coodinates is to appl a change of coodinates diectl to the D Catesian vesion qns...8 as otlined in the Appendi to this section Altenativel the eqations can be deived fom fist pinciples b consideing an element of mateial sbjected to stesses and as shown in Fig The dimensions of the element ae Δ in the adial diection and Δ (inne sface) and ( Δ) Δ (ote sface) in the tangential diection. Δ Δ Δ Δ Δ Fige 4..: an element of mateial Smming the foces in the adial diection leads to F Δ ( Δ) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ ( ) Δ ( ) Δ 0 (4..) Fo a small element and so dividing thogh b Δ Δ Δ ( Δ) 0 (4..) A simila calclation can be caied ot fo foces in the tangential diection { Poblem }. In the limit as Δ Δ 0 one then has the two-dimensional eqilibim eqations in pola coodinates: Solid Mechanics Pat II 60 Kell
2 Section 4. ( ) 0 0 qilibim qations (4..3) 4.. Stain Displacement Relations and Hooke s Law The two-dimensional stain-displacement elations can be deived fom fist pinciples b consideing line elements initiall ling in the and diections. Altenativel as detailed in the Appendi to this section 4..6 the can be deived diectl fom the Catesian vesion qns...5 -D Stain-Displacement pessions (4..4) The stess-stain elations in pola coodinates ae completel analogos to those in Catesian coodinates the aes thogh a small mateial element ae simpl labelled with diffeent lettes. Ths Hooke s law is now [ ] [ ] zz ( ) Hooke s Law (Plane Stess) (4..5a) Hooke s Law (Plane Stain) (4..5b) [( ) ] [ ( ) ] 4..3 Stess Fnction Relations In ode to solve poblems in pola coodinates g the stess fnction method qns. 3.. elating the stess components to the Ai stess fnction can be tansfomed g the elations in the Appendi to this section 4..6: φ φ φ φ φ φ (4..6) It can be veified that these eqations atomaticall satisf the eqilibim eqations 4..3 { Poblem }. Solid Mechanics Pat II 6 Kell
3 Section 4. Solid Mechanics Pat II Kell 6 The bihamonic eqation 3..3 becomes 0 φ (4..7) 4..4 The Compatibilit Relation The compatibilit elation epessed in pola coodinates is (see the Appendi to this section 4..6) 0 (4..8) 4..5 Poblems. Deive the eqilibim eqation 4..3b. Veif that the stess fnction elations 4..6 satisf the eqilibim eqations Veif that the stains as given b 4..4 satisf the compatibilit elations Appendi to 4. Fom Catesian Coodinates to Pola Coodinates To tansfom eqations fom Catesian to pola coodinates fist note the elations ) / actan( (4..9) Then the Catesian patial deivatives become (4..0) The second patial deivatives ae then
4 Section 4. Solid Mechanics Pat II Kell 63 (4..) Similal (4..) qilibim qations The Catesian stess components can be epessed in tems of pola components g the stess tansfomation fomlae Pat I qns Ug a negative otation (see Fig. 4..) one has ( ) (4..3) Appling these and 4..0 to the D Catesian eqilibim eqations 3..3a-b lead to ( ) ( ) 0 0 (4..4) which then give qns Fige 4..: otation of aes
5 Section 4. The Stain-Displacement Relations Noting that (4..5) the stains in pola coodinates can be obtained diectl fom qns...5: ( ) (4..6) One obtains simila epessions fo the stains and. Sbstitting the eslts into the stain tansfomation eqations Pat I qns ( ) (4..7) then leads to the eqations given above qns The Stess Stess Fnction Relations The stesses in pola coodinates ae elated to the stesses in Catesian coodinates thogh the stess tansfomation eqations (this time a positive otation; compae with qns and Fig. 4..) ( ) (4..8) Ug the Catesian stess stess fnction elations 3.. one has φ φ φ (4..9) and similal fo. Ug 4..- then leads to Solid Mechanics Pat II 64 Kell
6 Section 4. Solid Mechanics Pat II Kell 65 The Compatibilit Relation Beginning with the Catesian elation.3. each tem can be tansfomed g 4..- and the stain tansfomation elations fo eample ( ) (4..0) Afte some length calclations one aives at 4..8.
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