【正文】 ??sss 2222third (cubic) invariant（三次不變量） 3211 sss ???J? ?1332212J ssssss ????3213 sss?Jsij: nine ponents (sij =sji ) six ponents (principal surface) three ponents Algebraically 321 sss ??When the direction of the coordinate axes coincide with the principal direction 123000000ijssss????????? Some stress states in terms of principal stresses Uniaxial stress state（單向應(yīng)力狀態(tài)） , (+, 0, 0) uniaxial tension （單向拉伸） 01 ?s 032 ?? ss ( 0, 0, ) uniaxial pression（單向壓縮） 021 ?? ss 03 ?sPlane stress state (biaxial stress state)（平面應(yīng)力狀態(tài)） 01 ?s 02 ?s 03＝s (+,+,0) biaxial tension 01 ?s 02 ?s 03 ?s tension in one direction, pression in another direction. (+,0,) 01 ?s 02 ?s 03 ?s (0,,) biaxial pression Triaxial stress state (三向應(yīng)力狀態(tài) ) 01 ?s 02 ?s 03 ?s 321 sss ??(+,+,+) (,,) (+,,) (+,+,) Spherical stress state 01 ?s 032 ?? ss032 ?? ss 01 ?sDrawing of a round bar Extrusion of a round bar 132 sss ?? 321 sss ??Cylindrical stress state or 0 triaxial uniform tension 0 triaxial uniform pression (hydrostatic stress state). 321 sss ?? principal shear stress and maximum shear stress (主剪應(yīng)力和最大剪應(yīng)力 ) Relation between principal stress and principal shear stress The principal directions the coordinate axes Coincide YOZ: s1 ( sxx = s1 , ?xy = ?xz = 0) XOZ: s2 ( syy = s2 , ?yx = ?yz = 0) YOX: s3 ( szz = s3, ?zx = ? zy = 0) ??????????321000000ssslS x 1s? mS y 2s? nS z 3s?2232222212222 nmlSSSS zyxR sss ??????232221 nmlnSmSlSS zyxn sss ??????(A) ? ? 2232221223222221222 nmlnmlSSS nRs ssssss ???????? (B) jiij Sl ?s（主應(yīng)力和主剪應(yīng)力之間的關(guān)系） If ( l , m , n ) change, then Ss change Select l, m, n Ss extremum ( 極大， 極小 ) Two of l, m, n are independent The problem of finding extremum value of Ss with restriction condition 222 1 mln ??? Ss is the function of l and m ? ? 02 ???lS s? ? 02 ???mS s ? ? ? ? ? ? 02131232231 ??????? ????? ssssss mll? ? ? ? ? ? 021 32232231 ??????? ????? ssssss mlm(C) 1??nl = m = 0, m = n = 0, 1??ll = n = 0, 1??mSs attains its minimum value on the three principal planes Ss = 0 1s?nS0?sS2s?nS0?sS0?sS 3s?? nR SS(A) ? ? ? ? 411 ???? ?? nc oslc os ? ? 21 ???? mc o s From (C) we can get (s1 – s3)(12l2) = 0 22??l 22??n22??m 22??n ? ? ? ? 411 ???? ?? nc osmc os ? ? 21 ???? lc os 22??l? ? ? ? 411 ???? ?? lc osmc os ? ? 21 ???? nc o s321 ??? , is principal shear stresses ? ?2 1 312? s s?＝ －(B) ? ?3 1 212? s s?＝ －(B) ? ?1 2 312? s s?＝ －(B) m = 0 0l?l = 0 0m?n = 0 0m?22??m? ?m a x 1 312? s s? ? ? is maximum shear stress 1 2l2 = 0 Exercise Problem 1 The stress state is shown as in , Please determine the resultant stress SR, the ponents of the resultant, Sx, Sy, and Sz, the normal stress Sn and shear stress Ss on the oblique plane, when the three direction cosines of the oblique plane are x y z 10 5 10 5 5 5 13l m n? ? ? Exercise Problem 2 The four stress tensor are known as Ta, Tb, Tc and Td, please determine whether they belong to the same stress state or not? 30 0 00 20 00 0 10aT?????????30 0 00 15 50 5 15cT?????????20 0 00 20 00 0 20bT?????????25 5 05 25 00 0 10dT?????????Principal Stresses Definition: The oblique plane on which SS = 0 is a principal plane. The stress Sn = SR acting on the principal plane is a principal stress. The direction of the principal stress is a principal direction. Summary of Last Class ? ? 0?? jnijij lS?s0x x n y x z xij ij n x y y y n z yx z y x z z nSSSSs ? ?s ? ? s ?? ? s?? ? ? ??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? ? ? ? ?Stress invariants First (linear ) invariant Second (quadratic) invariant third (cubic) invariant 將 sij 代入 ( sij?ijSn )li= 0 的兩個(gè)方程中，并考慮到 l2 + m2 + n2 = 1 , 則可以得到三組方向余弦 l1, m1, n1。 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。 l2, m2, n2。 Substituting sij into two of the equations ( sij?ijSn )li= 0 and adding the condition l2 + m2 + n2 = 1 , the three set of direction cosines l1, m1, n1。 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