The elasticity and resistance of the materials of engineering . y point from OX^ measured parallel to ZY^ is called r\ thethird co-ordinate, normal to r and (p, is the co-ordinate x^ asbefore. It is important to observe that the co-ordinates ;r, rand <7?, at any point, are rectangular. The indefinitely small portion of material to be consideredwill, as shown in Fig. i, be limited by the edges dx, dr andr dcp. The faces dxdr are inclined to each other at the angled(p. Art. 7.] EQUATIONS IN SEMI-POLAR CO-ORDINATES. 21 The intensities of the normal stresses in the directions ofX and r will be
The elasticity and resistance of the materials of engineering . y point from OX^ measured parallel to ZY^ is called r\ thethird co-ordinate, normal to r and (p, is the co-ordinate x^ asbefore. It is important to observe that the co-ordinates ;r, rand <7?, at any point, are rectangular. The indefinitely small portion of material to be consideredwill, as shown in Fig. i, be limited by the edges dx, dr andr dcp. The faces dxdr are inclined to each other at the angled(p. Art. 7.] EQUATIONS IN SEMI-POLAR CO-ORDINATES. 21 The intensities of the normal stresses in the directions ofX and r will be indicated by N^ and R, respectively. Theremainder of the notationwill be of the same gen-eral character as that inthe preceding Article;i,e,, T^^ will represent ashear on the face dr. r dcpin the direction of r, whileN^^ is a normal stress, inthe direction of cp^ on theface dx dr. The strains or dis-placements, in the direc-tions of ,r, r and cp^ willbe represented by Uy pand w; consequently theunbalanced forces in those directions, per unit of mass, will be:. d^u d^ft - d^wm -7—, m —J— and in —^r-dt dt df (I) Those forces acfing on the faces hf, fe, and he, will be con-sidered negative; those acting on the other faces, positive. Forces acting in the direction of r.— R . r dq) dx, and ; 4- Rr dm dx 4- [ ^ , • dr = r —j-dr -\- Rdr)dq)dx,\ dr dr J — T^ydr dx, and ; 22 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 7. -f- T^r^r dx ?\—-^-- dcp, dr dx, — T^r . r dq) dr, and ; + T^r. r dcp dr -\ -^ dx . r dq) dr. On the face dr dx, nearest to ZOX, there acts the normal stress ( N^^dr dx -\ -^ dcp, dr dx\— N. Now N has a com-ponent acting parallel to the face/^ and toward OX, equal toN sin {dcp) z= iV^l—? = Ndcp, But the second term of this product will hold {dcpy, hence it will disappear, at the limit, inthe first derivative of Ndcp .*. Ndcp — N^^dcp dr dx. Sincethis force must be taken as acting toward OX, it acts with thenormal forces on Jif, and, consequentl
Size: 1563px × 1598px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., book, bookcentury1800, booksubjectbuildingmaterials, bookyear1883
