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 上产克大睾 SHANCBHAI JUAO TONO UNTVEEETTY Stress boundary condition Assume the forces acting on the unit area of the surface in the axis direction are px,py Pz px=ol+tm+tn py=txl+o,m+tan p.=txl+tmto.n Physical meaning:the relationship between the unit surface force and the stress components Institute of Forming Technology Equipment 22

Institute of Forming Technology & Equipment Stress Stress boundary condition 22            p l m n p l m n p l m n z xz yz z y xy y zy x x yx zx          Assume the forces acting on the unit area of the surface in the axis direction are px,,py, pz. Physical meaning: the relationship between the unit surface force and the stress components

Stress 上产克大睾 SHEAMGHAI DUD TONO UHTVEREETTY Example No.1 Question:For the slope plane with I=m=n,solve S,SSS and Solution:First,ox=10、0y=10、0z=0、z+ Txy-Tyx-5、tx=ta=5、tyz=ty-0(unit: 10 Mpa)。 10 ,=+mn=10. +5. 1 +5. 20 3 S,=t,1+o,m+t,n=5. 15 √3 3 =t1+7=m+on=5. 1 5 3 +0 /3 /3 3 S=VS,2+S,2+S2=5 26 3 Institute of Forming Technology Equipment 23

Institute of Forming Technology & Equipment Stress Example No. 1 23 Question: For the slope plane with l=m=n, solve S, Sx, Sy, Sz, n and n。 Solution: First, σx=10、 σ y=10、 σz=0、 τxy= τyx= 5、 txz=tzx=5、 tyz=tzy=0 (unit: Mpa)。                                 3 5 3 1 0 3 1 0 3 1 5 3 15 3 1 0 3 1 10 3 1 5 3 20 3 1 5 3 1 5 3 1 10 S l m n S l m n S l m n z xz yz z y xy y zy x x yx zx          3 26 5 2 2 2 S  Sx  Sy  Sz 

Stress 上清充通大￥ SHEAMGHAI DUD TONO UHTVEREETTY Example No.1 10 o =o P2+o,m2+o.n'+27lm+27mn+2tanl 140 =(10+10+25+2.5)。= 3 3 650 40 350 5 14 V93 Institute of Forming Technology Equipment 24

Institute of Forming Technology & Equipment Stress Example No. 1 24 3 40 3 1 (10 10 2 5 2 5) 2 2 2 2 2 2          l  m  n  lm  mn  nl n x y z xy yz zx        14 3 5 9 350 3 40 3 650 2 2 2       n  S  n  

Stress 上清充通大￥ SHEAMGRAI DUD TONO UHTVEREETTY Einstein's summation convention and tensor Einstein's summation convention Example 1:Multiplication of matrix and vector Ols+nly+tal i=x.y,= Eolx+ol,+tl: ∑ol o. 1.isls+inl,+ol: i=x,y,Z ∑ol =x,y,2 The right side of the equation can be written as: ∑o,ho rd i,j=x,y,z Institute of Forming Technology Equipment 25

Institute of Forming Technology & Equipment Stress Einstein's summation convention and tensor 25    xz yz z xy y zy x yx zx            zyxlll         xz x yz y z z xy x y y zy z x x yx y zx z l l l l l l l l l             i x y z iz i i x y z iy i i x y z ix i lll , , , , , ,  Einstein’s summation convention  Example 1: Multiplication of matrix and vector  i jx y z ij i l , , ,  ij i  l  The right side of the equation can be written as: or

Stress 上清充通大￥ SHEAMGRAI DUD TONO UHTVEREETTY Einstein's summation convention and tensor Free index Free index:the index j is called a free index.If one term has a +ob free index j,all terms must have it,to be consistent.One free index indicates n separate Dummy index Dummy index equations. Dummy index:the repeated index i is called a dummy index. Because it can be replaced with any other letter and the sum is the same.For example: a.-o写=o+a,+a) 3 k=x,y,zor1、2、3 Institute of Forming Technology Equipment 26

Institute of Forming Technology & Equipment Stress Einstein's summation convention and tensor 26  Dummy index: the repeated index i is called a dummy index. Because it can be replaced with any other letter and the sum is the same. For example:  ijai  mjbm Dummy index Dummy index Free index  n  ii 3 1  ijk xyz 、 、 ， ， or 、 、  1 2 3 ( ) 31  jj   x  y  z 31   kk 31   Free index: the index j is called a free index. If one term has a free index j, all terms must have it , to be consistent. One free index indicates n separate equations