The elasticity and resistance of the materials of engineering . nd the reasoning con-nected with the equation above, will give: dw _ drf p 00 tan ^ dz r cos tp dcp r r Without explanation there may at once be written: dv _ dpdy dr Fig. I of this, and Fig. 2 of the preceding Art. give : dti dco GO , dv dp 1 = -TT , and - dy dr r dx r dip These are to be used in the expression for T^r^ Preciselythe same Figs, and method give : dv __ dp _. dw _ drf tj ^ dz r cos ip dcp dj/ dr r which are to be used in finding T^r* 3 34 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 8. The expression for -^—7 will be
The elasticity and resistance of the materials of engineering . nd the reasoning con-nected with the equation above, will give: dw _ drf p 00 tan ^ dz r cos tp dcp r r Without explanation there may at once be written: dv _ dpdy dr Fig. I of this, and Fig. 2 of the preceding Art. give : dti dco GO , dv dp 1 = -TT , and - dy dr r dx r dip These are to be used in the expression for T^r^ Preciselythe same Figs, and method give : dv __ dp _. dw _ drf tj ^ dz r cos ip dcp dj/ dr r which are to be used in finding T^r* 3 34 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 8. The expression for -^—7 will be composed of the sum of two parts. In Fig. 2, ab is the original position of r dip, and afterthe strain rf exists it takes the position ec. Consequently ac (equal and parallel to bd and perpendicularto ak) represents the strain 7;, while cd rep-resents dq. Since, also, fc is perpendicu-lar to ck, the strains of the kind 7/ changethe right angle fck to the angle fee; orthe angle eck is equal to dw J , 1 1 ^d ea —rr = eed -\- dek r= —- _i- _-dx de ak /—. = -J- 4- ^r dip r eot tp In Fig. 2, the points a, b and k are iden-tical with the points similarly lettered in Fig. i. The expres- dusion for ?— may be at once written from Fig. i. There may, CI M then, finally be written : , dwdx drf Tf tan ip r dip . du doD and, -r-, = r— dz r eos ip dq) These equations will give the expression for T^^,The value of ^ _ die dv dwdx dy dz now takes the following form : drf dr r eos rp dcp dcD , 2p GO tan 0 . r dip r r - Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 35 The last two terms are characteristic of the spherical co-ordinates. The equations (20), (21), (22), (11), (12) and (13), of Art. (5),take the forms: N. =^^e + 2g^4 (5) N, = ^^ e + 2g( ^y , + ^ - 2L^^^\ (6) I — 2r \r cos ^ dcp r r J ^ ^ N,=^^e + 2G(^+!^) (7) ^ I — 2r \r dip rj 7;,= c(^+ _i^+^i^). ... (8) \r dip r cos ^ dcp r J „ ^fdoD GO dp \ , X T^=g(^-^+^JI-^) (10) \r cos f dcp dr rJ If these values are inserted In E
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