1、Simulations of plastic processing:elasto-plastic deformationPart 1:tensors and continum mechanicsTeacher:法法 QQ号：2306847727，office 423-2 A区Introduction In this class,we study materials at a macroscopic scale(宏观尺度):between dislocation(位错)and sample(样品)We use mathematics,mechanics,materials propertiesH
2、ongguan chiduPlan(2 weeks,6 classes)Part 1:tensors and continuum mechanics张量和连续介质力学 Part 2:elastic deformation弹性变形 Part 3:plastic deformation塑性变形Zhangliang he lianxvjiezhi lixueSuxing bianxingtanxing bianxingContent Material element 材料单元 Stress tensor 应力张量 Strain tensor 应变张量Yingli zhangliang Several
3、 years ago,you have learn Newtonian mechanic:the solids were undeformable,and sometime modeled as material point.In material science,we study the mechanical behavior(力学行为)inside a deformable body of material body of material=sample,beam,mechanic piece,bridge Material elementLixue xingweiMaterial ele
4、ment The material is a continuum(连续体)i.e.the matter is continuously distributed in the body(no void,no cracks(无空位，无裂纹)So the body can be sub-divided into infinitesimal elements(无穷小的单位元)Wu kongweiWu liewenWu qiongxiao de dan weiyuanlianxvtiMaterial element The body of material is divided into materia
5、l elements These material elements are also called infinitesimal elements(for derivation /x)or representative volume elements Each material element can have different properties（性能）and mechanical response(stress,strain)Material elementMaterial=body=solidcut the material with a virtual plane(虚拟的平面)Ma
6、terial element (1 face is in the plane)X0YZBasisxvni How to define 1 element:Material elementWe want to study in 3 dimensions,so use 3 planes defined with the basis:X0YZBasisPlane normal to YPlane normal to ZPlane normal to XMaterial elementdxX0YZdydzWe study the behavior of 1 element of the materia
7、l,of dimension 111.element size has no unit,but the size of the element much bigger than the atoms(原子),to be a continuum.The size depend on what we want to study,of what accuracy we need.YuanziMaterial element In mechanic,the material element contain many grains In crystal plasticity(晶体的塑性),the mate
8、rial element is smaller than a grain In continuous materials,the forces are related to the bodys deformation through constitutive equations(本构方程)The internal forces are continuous and the material displacement is a derivable function(可导的函数)Jingti de suxingBengou fangchengKedao de hanshuStress tensor
9、 We saw that the body of material can be subdivided into material elements We want to study what happen to the body of material when some forces are applied to the body So we must study what happen to each element of the material We study how the forces are transmitted to the elementsStress tensorHy
10、pothesis on applied forces:The solid is in equilibrium(do not move)Some forces P are applied on the solid Equilibrium of forces and moment of forces/Torques(应力与动量平衡)P1P2P4P3P1+P2+P3+P4=0Equilibrium in rotation:M1+M2+M3+M4=0(Mi=PiOA,Pi applied on A)X0YZYingli yu dongliang pinghengStress tensorOrigin
11、of the stress tensor:solid and internal stress We cut the solid(body of material)along a planeP1P2P4P3X0YZVirtual plane to cut the materialVirtual surface inside the materialStress tensor Fr is the force applied on the left part by the right part External forces can be applied on points,surfaces,or
12、volume;but internal forces are surface forces.The forces Fr and Fl are equivalent on the 2 sides of the solidP1P2P4P3FlrlFrStress tensor The force Fr=-(P1+P2)is discomposed into a normal component Fn(垂直分量)and a shear component Ft(剪切分量)in the plane(tangential force)In compression,the normal force Fn
13、is negativeFrl-FrlP1P2FnFtChuizhi fenliangJianqie fenliangStress tensor On the surface dS of the element,we have the force per unit surface r=stress(应力)The force on the right face of the element dS=dzdy is:S rdS=yz rdydz=-Fr-FrlFnFtrntlX0YZdSStress tensor If the internal force is homogeneous(均匀的)on
14、the surface S,F=S Usually the stress distribution in the material is not homogeneous For complex cases,we may use Finite Elements programs(有限元软件)to solve the constitutive equations(for example the Hooke law)and compute the stress(and strain)in all the elements.JunyunYouxianyuan ruanjianStress tensor
15、 We apply the virtual cut analysis on the 3 faces of the material element:we get 3 vector of stress=force/surfacertfStress tensor We discompose t,r,and f,into 3 component in the basis:we get 9 components for the stresses.yzzzzyxzxyxxzxyyyx0XZYrtfStress tensor Normal stresses(正应力)ii,i=x,y,z Shear str
16、esses(剪切应力)ij,i,j=x,y,z,ij xy is the stress component in the direction x on the plane of normal y.yzzzzyxzxyxxzxyyyx0XZYStress tensor The stress tensor is used to compute the force F on a face of normal n(and surface S=1):F(n)=n=So is the force on the face normal to X:r=-Frxx xy xz yx yy yz zx zy zz
17、nx ny nzxx yx zxzyxyyyStress tensor The Stress tensor on a material element depend on the position=(x,y,z)is a function of the position x,y,z of the element Stresses on faces Y：0XZYzy+zy/y*dyxy+xy/y*dyyy+yy/y*dyStress tensor Stresses on faces X and Z In continuum mechanics,the stresses are continuou
18、s.Finite Element methods use equilibrium principles to compute heterogeneous(不均匀的)stresses and deformations zx+zx/x*dxxx+xx/x*dxyx+yx/x*dxzz+zz/z*dzxz+xz/z*dzyz+yz/z*dz0XZYStress tensor The stresses through the material element must be in equilibrium:Equilibrium in forces:div()=0 Equilibrium in mome
19、nt of forces(rotation):=T the tensor is symmetric(ij=ji)xx/x+xy/y+xz/z=0yx/x+yy/y+yz/z=0zx/x+zy/y+zz/z=0Stress tensor Whatever is the stress tensor,there is a basis in which there is no shear:=In this basis,the stress are called Principal stresses0XZY0e1e3e2I 0 0 0 II 0 0 0 IIIxx xy xz yx yy yz zx z
20、y zzStress tensor Change of basis(=referential,coordinate system):simple shear=plane stress0XZY0e1e3Y=e2zx=10MPaxz=10MPa11=10MPa33=-10MPaXYZ=0 0 10 0 0 0 10 0 0123=10 0 0 0 0 0 0 0 -10Demonstration:Express a matrix into another basis Transformation Matrix from(XYZ)to(e1 e2 e3):T=(express ei inside X
21、YZ)123=T-1xyzT1/20-1/20101/201/2Stress tensor Pressure p(球应力):p=Stress deviator tensor(应力偏张量)s:s=pId=Deviatoric of =s11+s22+s33=0 The stress deviator tensor is useful because pressure do not produce plastic deformationxx+yy+zz 3xx-p xy xz yx yy-p yz zx zy zz-pying li pian zhang liangQiuyingliStress
22、tensor Equivalent von Mises stress(等效应力)=VM=3/2(sxx2+syy2+szz2+2sxy2+2sxz2+2syz2)Also called Equivalent tensile stress because in uniaxial tension p=xx/3:VM=3/2(xx-p)2+p2+p2)=3/2(4/9+1/9+1/9)xx =xx Use to compare different stress states(应力状态)Use with the von Mises equivalent strain:VM=2/3(xx2+yy2+zz
23、2+2xy2+2xz2+2yz2)Dengxiao yingliYingli zhuangtaiStress tensor Von Mises stress in plane stresses(平面应力)p=(x+y)/3 =VM=3/2(x-p)2+(y-p)2-p2)=x2+y2-xy Draw VM=constantxxyyxyVM=x=yx=-y=VM/3Stress tensor Von Mises stress in principal stress basis(i.e.if ij,ij=0)The constant von Mises stress is a cylinder(圆
24、柱体)tilted at 45 off the axisyxz(111)Yuan zhu tiStress tensor Compute and as a function of Mohr circle in plane stress is maximum for=45xxyy2xxyy=xsin2+ycos2=ysincos-xcossinchange of basis for Mohr circle T=xyz=123=T-1xyzT =sin-cos0cossin0001x000y0000 xsin2+ycos2 ysincos-xcossin 0ysincos-xcossin xcos
25、2+ysin2 0 0 0 0Strain tensor We saw the body of materials and the material elements are submitted to forces.In material science,as in the real life,the materials can deform We need to measure and quantify these deformations Usually these deformations are very smallStrain tensor Origin of the strain
26、tensor:the displacement of material Displacement field(位移场)u(x,y,z)for each point X(x,y,z)of the material u=eyezbeforeafteru(x,y,z)0X(x,y,z)exux uy uzWei yi changStrain tensor Displacement gradient tensor(位移梯度张量)G=grad(u)=The symmetric(对称的)part of G is=1/2(G+GT)is the strain tensor(or deformation te
27、nsor)The anti-symmetric(非对称的)part of G is=1/2(G-GT)is the rotation tensor(or spin tensor)ux/x ux/y ux/z uy/x uy/y uy/z uz/x uz/y uz/zWei yi ti du zhangliangFei duicheng deStrain tensor=ux 1 ux uy 1 ux uz x 2 y x 2 z x 1 uy ux uy 1 uy uz 2 x y y 2 z y1 uz ux 1 uz uy uz2 x z 2 y z z (+)(+)(+)(+)(+)(+)
28、Strain tensor Principal strain:as the strain tensor is symmetric,there is a coordinate system(basis，坐标系)where ij=ij=0 In this basis xx,yy and zz are call the principal strain(eigenvalues of the matrix，特征值)=YXe1e2Zuo biao xiTezheng zhiStrain tensorExamples of strain tensor:Pure shear u=ux=yG=Plane st
29、rain tension u=0 /2 0/2 0 0 0 0 0-0 0 0 0 0 0 0YX(x-a)(y-b)0e2e1(a,b,0)sin()tan()(in radian)0 0 0 0 0 0 0 0 Strain along the elongation direction y for homogeneous deformation:uy=(lf-l0)/l0y Gyy=(lf-l0)/l0=yyStrain tensoryxl0lfuy y Total strain along a uniform elongation path:Current length l at t,l
30、+dl at t+dt=dl/l is the elongation during dt total=lilfdl/l=ln(l)lilf=ln(lf)-ln(li)=ln()Strain tensorlf lillilfStrain tensor Change of volume(2D):V=dxdydz dV=(dx+xxdx)(dy+yydy)dz-dxdydz/dxdydz xx+yy (we neglect xxyy)dxdyxxdxyydyxxdydzxxdydzStrain tensor Uniform deformation(均匀变形):the strain tensor is constant through the material(do not depend on(x,y,z)is the sum of the elastic and plastic strain In plastic deformation,the volume is constant(体积不变),so xx+yy+zz=0 Shear does not produce any change of volume.
