上海交通大学：《材料加工原理 Principles of Materials Processing》教学资源（课程讲稿）03. Technology of plasticity——Elements of plastic theory

Stress 上清充通大￥ SHEAMGHAI DUD TONO UHTVEREETTY The stresses acting on arbitrary plane Assume:differential diagonal plane's area>ds; I,m,n>cosines of angel 1=cos(N,X)m=cos(N,Y)n=cos(N,Z) On > So,three vertical coordinate plane's area is as follows: SoBc SABc cos(n,x)=ds.1 Soc SABc cos(n,y)=ds.m SoB4 SaBc cos(n,2)=ds.n A Direction cosine satisfy 12+m2+n2=1 Institute of Forming Technology Equipment 17

Institute of Forming Technology & Equipment Stress The stresses acting on arbitrary plane 17 Assume: differenƟal diagonal plane’s area → ds; l, m, n → cosines of angel l NX  cos( , ) m NY  cos( , ) n NZ  cos( , ) S S   n x ds l OBC ABC  cos ,   SOAC  S ABC cos  n, y  ds m SOBA  S ABC cos  n,z  ds n 1 2 2 2 l  m  n  z x y o x  y z  xy  yz  yx  xz zy zx Sn n Snx Sny Snz n  n B A  So, three vertical coordinate C plane’s area is as follows:  Direction cosine satisfy

Stress 上产克大睾 SHEAMGRAI DUAD TONO UNTVEEETTY The stresses acting on arbitrary plane Since the free body must be in equilibrium,the forces acting on each of its faces must balance. ∑X=0 ∑Y=0 ∑Z=0 In the x direction Sm·d-ox·dl-tw·dsm-tx·dsn=0 In the y direction Sds-tsds.1-o'ds.m-tsds.n=0 In the z direction Sds-tx=ds.I-Tyds.m-o:.ds.n=0 Institute of Forming Technology Equipment 18

Institute of Forming Technology & Equipment Stress The stresses acting on arbitrary plane 18 X  0 Y  0 Z  0 Snx  ds  x  dsl  yx  ds m  zx  ds n  0 S  ds   dsl   dsm   ds n  0 ny xy y zy    Snz  ds  xz  dsl  yz  dsm  z  ds n  0 In the x direction In the y direction In the z direction Since the free body must be in equilibrium, the forces acting on each of its faces must balance

Stress 上清充通大￥ SANCEAI JUO TO阳UY The stresses acting on arbitrary plane >After finishing the equation Sm=xl+Tmm+tan Sw=tol+o,m+tn Sm=Txl+tym+o.n Expressed as a matrix x S ×m S n z Institute of Forming Technology Equipment 19

Institute of Forming Technology & Equipment Stress The stresses acting on arbitrary plane 19            S l m n S l m n S l m n nz xz yz z ny xy y zy nx x yx zx                nml SSS xz yz z xy y zy x yx zx nz nynx           After finishing the equation  Expressed as a matrix

Stress 上清充通大￥ SHEAMGRAI DUD TONO UHTVEREETTY Normal stess and shear stress Total stress: S%=S2+S?+S2 Ox Normal stress: ty/ S On=Sml+Smm+Srn T o,=G12+o,m2+o.n2+27s lm+2rmn+2tanl Shear stress: Y0: =S -OR Institute of Forming Technology Equipment 20

Institute of Forming Technology & Equipment Stress Normal stess and shear stress 20 Total stress: Normal stress: Shear stress: 2 222 n xyz SSSS  2 2 2 n Sn  n    S l S m S n  n  nx  ny  nz l m n lm mn nl n x y z xy yz zx     2 2 2 2 2 2      

Stress 上清充通大￥ SHEAMGRAI DUAD TONO UNTVEEETTY The stresses acting on arbitrary plane >If the axis is the principal axis,The shear stress on the cross section perpendicular to the coordinate axis is zero. G,=012+G,m2+o:n2+27xlm+27mn+2tnl Simplify to 0n=0,2+02m2+0,n2 Since, S,=0p+oim2+oin S0, x=(g1-o2}12m2+(o2-o3}m2n2+(o3-o1}n212 Institute of Forming Technology Equipment 21

Institute of Forming Technology & Equipment Stress The stresses acting on arbitrary plane 21 l m n lm mn nl n x y z xy yz zx     2 2 2 2 2 2       2 3 2 2 2  n 1l  m  n 2 2 3 2 2 2 2 2 Sn 1 l  m  n      2 2 2 3 1 2 2 2 2 3 2 2 2 1 2 2 l m m n n l  n           If the axis is the principal axis, The shear stress on the cross section perpendicular to the coordinate axis is zero. Simplify to Since, So