【正文】 l2, m2, n2。 Which can be written as: Where The cubic equation（三次方程） has three real roots which are the three principal stresses,s1 ,s2 and s3 acting on three orthogonal plane. 321 2 3 0n n nS J S J S J? ? ? ?1 x x y y zzJ s s s? ? ?? ? ? ?2222 x x y y y y zz zz x x x y y z zxJ s s s s s s ? ? ?? ? ? ? ? ? ?? ?2223 2x x y y z z x y y z z x y z x x z x y y x y z zJ s s s ? ? ? ? s ? s ? s? ? ? ? ?三次方程有三個(gè)實(shí)數(shù)根，這三個(gè)根分別為作用在三個(gè)相互垂直的坐標(biāo)面上的主應(yīng)力 s s2 和 s3。r Angewandte Mathematik und Mechanik”, and led the area of applied mechanics. He is also famous as the teacher of many leaders in mechanics in the 20th century such as Th. von K225。vy Brief historical account When Tresca presented his paper to the French Academy of Sciences, Barr233。 Three dimensional stress analysis (important feature of tensor) Resultant stress on an oblique plane inclined to three Cartesian axes Take an element using the method of sections Intersected by Three Cartesian plane Tetrahedron OABC (四面體 ) x z y O A B C xsxz?xy?ys yz?yx?zszx?zy?RSnS sSzSxSySl m n RSzSySxS222R x y zS S S S? ? ?An oblique plane ABC （三維應(yīng)力分析） （與三個(gè)坐標(biāo)軸相傾斜的斜面上的和應(yīng)力） Determine according to on three Cartesian planes: RS ijsplane area ox oy oz OBC l sxx ?xy ?xz OAC m ?yx syy ?yz OAB n ?zx ?zy szz ABC 1 Sx Sy Sz OABC in equilibrium 000XYZ? ?? ??? ?????x x x y x zxS l m ns ? ?? ? ?From first column of the table y x y y y zyS l m n? s ?? ? ?From second column of the table z x z y z zzS l m n? ? s? ? ? From third column of the table ??????????????????????nmlllllzyxi iijj lS s? j x j x y j y z j zx j y j z jS l l ll m ns s ss s s? ? ?? ? ?from sub. j=x x x x y x zxS l m ns ? ?? ? ? j=y y x y y y zyS l m n? s ?? ? ?j=z z x z y z zzS l m n? ? s? ? ?222R x y zS S S S? ? ?ijsilis known is known Sj is known SR Therefore, if the stresses on three orthogonal Cartesian planes are known, the stresses on any oblique plane can be determined. ijs can be transformed ji ??s Normal stresses on the oblique plane （斜面上的正應(yīng)力） SR ? ? ? ? ? ? ? ? r r r r oR SSSS ??? ? ? ? ? ? ?n x x x y x z y x y y y z z x z y z zS l m n l l m n m l m n ns ? ? ? s ? ? ? s? ? ? ? ? ? ? ? ?? ?2 2 2 2x y z x y y z z xl m n lm m n n ls s s ? ? ?? ? ? ? ? ? Sn normal ponent, is coincide with ON Ss shear ponent resolve shear stresses on the oblique plane （斜面上的剪應(yīng)力） 222 snR SSS ?? 22s R nS S S?? Stresses boundary conditions （應(yīng)力邊界條件） Relations between the distribution load on the body surface and stresses within the body at the same boundary point are the stress boundary condition. （所謂應(yīng)力邊界條件是指物體表面上的應(yīng)力分布與同一邊界處物體內(nèi)部應(yīng)力之間的關(guān)系。 l2, m2, n2。 Spherical stress tensor Hydrostatic pressure : mp s??mijmmms?sss???????????000000ms Spherical stress state ????????????????ppppijmij000000?s? In general in metal forming process ms 0 Deviator stress and deviator stress tensor （偏差應(yīng)力和偏差應(yīng)力張量） Any arbitrary （任意的） stress state ijs ???????????????????????????????????????mmmzzyzxyzyyxxzxyxzzyzxyzyyxxzxyxmijijijssss???s???ss???s???ss?ss000000mijijij s?ss ???mijs?Spherical stress tensor volumetric ponent of deformation mijijij s?ss ???(Deviator stress tensor distortional . of deformation) resolve ijs?????is deviator stress tensor which produce distortional ponent of deformation ijs?????被定義為偏差應(yīng)力張量，該張量使得物體產(chǎn)生形狀變化。 l3, m3, n3。） Stress ponents (應(yīng)力分量 ) F?rs, and also, needs not be normal to the section plane. F?resolve （分解） sF?NF?: Normal to the reference plane : tangential to the reference plane NF?sF?F?A?PNormal stress (正應(yīng)力 ) NNA0F d FA d Alim??s????(Normal to the section plane) Positive: elongation Negative: pression Shear stress (剪應(yīng)力 ) ssA0F d FA d Alim???????(tangential to the section plane) Change in shape （改變物體的形狀） Nine ponents and stress tensor (九個(gè)分量和應(yīng)力張量 ) Coordinate system: Oxyz Take an infinite small element from the body around point P. Six section planes parallel to the coordinate planes Parallelpiped (平行六面體 ) Three orthogonal planes: xoy, yoz and zox On the face parallel to the plane xoy ( normal direction is oz ) Normal stress: z zzss? (In the oz direction) Shear stress: z?resolve zx?zy?Along ox direction Along oy direction Double subscript notation First subscript: The direction of the normal to the plane on which the stress acts. second subscript: The sense of the stress . z y x O zszx?zy?xs xy?xz? ysyx?yz?yoz zox xoy ox oy oz yysxxszzsxy?yx?xz?yz?zy?zx?plane normal direction Sense of stress ox oy oz Another specified coordinate system oxyz? ? ?Another ponent system yys??xxs??zzs??xy???yx???xz???yz???zy???zx???They can be transformed Coordinate system infinite ponent system infinite Stress tensor Determine the stress state at the point Expression of stress tensor ( matrix of tensor) －－應(yīng)力張量的表示方法（張量的矩陣形式） x x x y x zy x y y y zz x z y z zs s ss s ss s s????????x x y x zy x y y zzx zy zs ? ?? s ?? ? s????????x x x y x zy x y y y zz x z y z zs ? ?? s ?? ? s????????( , , , )ij i j x y zs ?Row Act on the same plane, but in the different direction Columm Act on the different plane, but in the same direction 1. The nine