【導(dǎo)讀】塑性理論主要考慮塑性變形體內(nèi)應(yīng)力－應(yīng)變的數(shù)學(xué)關(guān)系。

　　

【正文】 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。 l2, m2, n2。 l3, m3, n3。 這三組方向余弦便確定了三個(gè)相互垂直的主方向 . 3211 sss ???J? ?1332212J ssssss ????3213 sss?JWhen the direction of the coordinate axes coincide with the principal direction 123000000ijssss?????????Algebraically 321 sss ??Some stress states in terms of principal stresses Uniaxial stress state（單向應(yīng)力狀態(tài)） Plane stress state (biaxial stress state)（平面應(yīng)力狀態(tài)） Triaxial stress state (三向應(yīng)力狀態(tài) ) ? ?2 1 312? s s?＝ －? ?3 1 212? s s?＝ －? ?1 2 312? s s?＝ －? ?m a x 1 312? s s? ? ? is maximum shear stress principal shear stress and maximum shear stress 3 The stress tensors of the body is known as sij, Please determine the principal stress s1 , s2 and s3, the direction cosine of the principal plane , and the principal shear stress. 1 0 10 1 01 0 1ijs??????????????Exercise Problem 4 The plane stress state is shown as in Fig. 3, Please verify that : 22c os 2 sin c os sinn x x y ys s ? ? ? ? s ?? ? ?? ? ? ?22sin c o s c o s sinn x y x y? s s ? ? ? ? ?? ? ? ?? ? ? ?11 c o s 2 si n 222n x y x y x ys s s s s ? ? ?? ? ? ? ?? ?1 si n 2 c o s 22n x y x y? s s ? ? ?? ? ?or xsysxy?yx?nsn?Fig. 3 ?Exercise Problem 5 在 oxy平面內(nèi)按圖 4所示的方向貼應(yīng)變片，測(cè)出的應(yīng)力為 ,x y ns s s試求 ?xy? ? ?n? ?x y n sn sx sy 45o Exercise Problem Spherical and deviator stresses (球應(yīng)力和偏差應(yīng)力 ) Deformation Volumetric ponent: elastic, volumetric change (hydrostatic ponent) Distortional ponent: change in geometric of a body, (elastic or plastic). ijsvolumetric ponent (elastic) Spherical stress Deviator stress distortional ponent (elastic or plastic) Spherical stress and spherical stress tensor（球應(yīng)力和球應(yīng)力張量） Spherical stress state Spherical stress state is a uniform triaxial (tensile or pressive) stress state （球應(yīng)力狀態(tài)是均勻的三向應(yīng)力狀態(tài)（拉伸或壓縮）） Features: 1)shear stress is absent on any arbitrary plane 2)No any distortional ponent of deformation 3)give rise to volumetric change Spherical stress Consider the stress state ijor, ssss 321Define spherical or hydrostatic stress, or average stress （定義球應(yīng)力或靜水壓力，或者稱為平均應(yīng)力） ? ? ? ?1 2 3 11 1 13 3 3m x y z Js s s s s s s? ? ? ? ? ? ?or in tensor notation （用張量形式表示 ） 13m k kss? So sm is an invariant and independent of the choice of coordinate system. It produces volumetric change 所以 sm 是不變量，與坐標(biāo)系的選擇無關(guān)，它只能導(dǎo)致體積變化。 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。 Invariants of deviator stress tensor （ 偏差應(yīng)力張力不變量 ） ijs?tensor ijs?has three invariants as the stress tensor ijs03211 ?????????????? zyxJ ssssss3 0x m y m zxy xz y z mms s s ss s s ss s s s s? ? ? ? ? ?? ? ?? ? ? ??? ? ? ? ? ?