材料成型双语Chapter_6Stressevaluationapproach

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,6 Stress evaluation approach (slab method),(,应力分析方法工程法,),6.1 General consideration force and energy,Assumption,1,Plane section remain plane.,Element cut from a plastic deformed material with sections parallel to the plane section.,slab,Slab method,Equilibrium equations,Reduced from the equilibrium of the slab,The relation of equilibrium of stresses,(1),Assumption,2,Take some special directions as the principal direction.,Simplified yielding criterion.,For example,:,For plane strain,:,Horizontal,(,width,),. ox,vertical,(,thickness,),.,oy,y. c.,s,1,s,3,=,2,k =K,s,x,s,y,Yield stress in plane strain,For,axisymmetrical,problem,:,Axis of a work-piece oz,Radial direction . or,s,1,s,3,=,2,k =Y,s,z,s,r,(,Uniaxial,yield stress,),(2),(1) + (2),Distribution of tooling pressure,(,integration constants are included in the equations,),Stress boundary condition,:,for determining integration constants,.,Assumption,3:,Friction condition,1),Sliding friction, coulomb friction law: t =,m,p,(,m,= 0 , frictionless,),2),Sticking friction,(,常摩擦应力区,),(,粘着摩擦,),滑动摩擦,遵守库仑摩擦定律,(,常摩擦系数区,),t =k,(,in hot working, rough tooling surface,),3),Constant friction law,（摩擦应力递减区）,t =,mk,(0md,t,=,m,p k,When xd,t,= k= K/2,When x=d,m,p= K/2,Determine C,At x = w,Yield criterion,When x=d,m,p= k=K/2,When xd,t,= k= K/2,When x=d,m,p= K/2,t =,m,p,t = K/,2,t,xy,p,x,d,w,2T,6.3,Quasi- static plane strain drawing of sheet material through a wedge shaped die,1),Description of the process,Small semi-angle,:,a,t,= m,p,(,lubricate well,),s, longitudinal tensile stress from back tension,Problem,:,Drawing force t=,?,a,h,H,s,t,o,x,2),Take element and analyze the stress,Assumption,:,vertical section remain plane.,Slab cut with vertical section,Normal stress: Uniform,Shear stress: no.,x , x +,dx,x,a,h,H,s,t,o,y,Exit,Entrance,Stress analysis,Vertical section,x ,s,x,x +,dx,s,x,+d,s,x,Top and bottom die-,workpiece,interface:,p ,t,=,m,p,Vertical component,Horizontal component,Uniform on a section and there is no shear stress,Equilibrium equations.,S,x =,0 (,in horizontal direction,),x,s,x,dx,s,x,+,d,s,x,p,m,p,ds,Neglect the second order of the term,dx,ds,dy,a,The stresses in the vertical direction,dx,p,m,p,s,y,ds,Because,m,1,and,a,is very small.,Therefore,(A),3),The assumption of the principal direction,Principal direction,ox,oy,s,x,s,1,s,y,s,3,-p,Yield criterion,s,1,s,3,=,2,k = K,s,x,(,- p,),= K,s,x,= K- p,p,= K-,s,x,(B),4),Drawing force at the exit,(B),(A),Let,(C),Integrate,(C),(D),Determine the integrate constant,A,by means of boundary condition .,At entrance,y = H,s,x,= s,(D),(D),At exit,y = h,s,x,= t,5),Maximum fractional reduction in a single pass,（单道次最大压下率）,If,t,attains the yield stress in,uniaxial,tension (leaving the die), then the sheet yields. The limit condition is,When back tension is employed,If back tension is not employed,The maximum fractional reduction,For example.,m,=,0.05,.,a,=,10,o,.,cot,a,=,5.6713,.,B,=,m,cot,a,=,0.2836,.,(1+,B,) /,B =,4.53,.,Therefore .,Hence .,Thus .,6),Normal pressure on the die face,（模具表面的正压力）,At entrance,p,= K- s,At exit,7),Compression with inclined platens,a,h,H,t,1,o,t,2,w,2,w,1,Neutral plane,On the left side of the neutral plane,s,x,s,x,+,d,s,x,p,m,p,p,m,p,Yield criterion,s,x,(,- p,),= K,s,x,= K- p,d,s,x,= -,dp,s.b.c,. x = -w,2,s,x,= t,2,p = K t,2,a,h,H,t,1,o,t,2,w,2,w,1,Neutral plane,On the right side of the neutral plane,s,x,s,x,+,d,s,x,p,m,p,p,m,p,Yield criterion,s,x,(,- p,),= K,s,x,= K- p,d,s,x,= -,dp,s.b.c,. x = w,1,s,x,= t,1,p = K t,1,Flow stress depends on strain rate,(,strain rate sensitive,),High friction coefficient,.,6.4,Quasi- static plane strain cold rolling of plate and strip metal,1),Description of the process,hot rolling,stiction,cold rolling,Flow stress is independent of strain rate,Low friction coefficient,.,Coulomb law,section material,plate, strip,elongation - rolling direction,spread - transverse direction,compression- vertical direction,3,- Dimensional flow,elongation,spread ,10,The velocity of strip (,v,s,) in passing through the roll gap,at the entry,v,s,= v ,v,r,cos,f,.,Entry,neutral plane,backward zone,work-piece moves backward when referring to the rolls,neutral plane,exit,forward zone,work-piece moves forward when referring to the rolls,The relative velocity between rolls and strip,Direction of friction stress,Towards to the neutral point,From the incompressibility,Hbv,H,=,h,x,bv,x,=,hbv,h,Roll flattening,Assume :,The arc of contact is circular and of radius R.,RR,(,R is the radius of the un-deformed rolls,),Strain hardening,k,k,1,k,2,entry,exit,Tension - tensile stresses,The relation between,h,x,and,f,x,2),Distribution of normal roll pressure,Assume,:,Vertical plane section remain plane.,Take a slab,No shear stress on the sections and normal pressure is uniform,Stress analysis,:,f,f,+,d,f,Section,f : s,x,Section,f,+,d,f,:,s,x,+,d,s,x,Vertical section,y,(,h,x,),y+dy,(,h,x,+dhx,),Top and bottom interface,Pressure,p,:,normal to the interface.,Friction shear stress,t=,m,p,:,tangential to the interface.,Entry plane,exit,p,m,p,f,d,f,s,x,s,x,+,d,s,x,h,H,t,2,t,1,R,x,dx,p,m,p,s,x,s,x,+,d,s,x,m,p,p,In forward zone,In backward zone,p,m,p,s,x,s,x,+,d,s,x,m,p,p,y,dy/,2,Equilibrium equations,:,p,m,p,s,x,s,x,+,d,s,x,m,p,p,In forward zone,In backward zone,p,m,p,s,x,s,x,+,d,s,x,m,p,p,(A),(B),(C),T.Karman,equation,（卡尔曼方程）,1925,The stresses in the vertical direction,Because,Therefore,The assumption of the principal direction,Rolling direction,ox,oy,s,x,s,1,s,y,s,3,-p,Yield criterion,s,1,s,3,=,2,k,s,x,(,- p,),=,2,k,s,x,=,2,k- p,d,s,x,= d,(2,k- p,),(D),Vertical direction,(D),(C),Sub.,P253,Integrate,Let,The positive sign applies to the forward zone and the negative sign applies to the backward zone.,(E),Determine the integral constant according to the stress boundary condition,At exit,f,=,0,Q =,0,s,x,= t,2,p,+,=,2,k,2, t,2,y = h,k = k,2,From,(,E,),Sub.,(,E,),At entry,f,=,f,1,Q = Q,1,s,x,= t,1,p,-,=,2,k,1, t,1,y = H,k = k,1,From,(,E,),Sub.,(,E,),Position of the neutral plane,At neutral plane,f,=,f,n,Q =,Q,n,y =,h,n,p,+,= p,Since,6.5,M. D. Stone Cold Rolling Equation,Assumption,:,1),Simplify it as plane platens compression,2),Neglect wide spreading,3),Coulomb friction,t,f,=,m,p,4),Plane section remain plane,dx,s,x,s,x,+,d,s,x,p,m,p,s,f,s,b,l /,2,l /,2,In forward slip zone, the equilibrium equation is,In backward slip zone, the equilibrium equation is,Because,t,f,=,m,p,s,x,(,p,),=,2,k,d,s,x,=,dp,Therefore,Forward zone “+”,Backward zone “,”,In forward slip zone,Determine the integration constant,C,1,by means of stress boundary condition,When x =,-,l /,2,s,x,=,s,f,p = K,-,s,x,= K,s,f,=,K,(1,-,s,f,/,K,),In backward slip zone,Determine the integration constant,C,2,by means of stress boundary condition,When x = l /,2,s,x,=,s,b,p = K,-,s,x,= K,s,b,=,K,(1,-,s,b,/,K,),Rolling force P,Let,6.6,compression of thick plate with plane platens,(,l / h ,1),0,0.5,1.0,1.5,9.8,19.6,29.4,39.2,h,l,Without over end,h,l,With over end,When without over end, the compression force increase with the increase of,l/h,.,When with over end,If,l/h,1,The compression force is the same as that of without over end.,有外端,无外端,Assumption,On the interface of platens and work-piece, the lubrication is very well, so that,m,= 0,t,f,= 0,On the interface between contact area and over end zone,t,xy,=,t,e,= k = K/,2,Along x direction,t,xy,is linear and in-dependent of y,From the equilibrium equations,(a),V-M yield criterion,(b),h,l,t,e,=k,p,x,y,Differentiate the first equation of (a) to y, and second equation to x.,(c),According to (b),Because,s,x,and,s,y,are compressing force, and,therefore,(d),Sub,. (d),into,(c),(e),because,t,xy,is not related with y , therefore,Determine c,1,and c,2,according to the stress boundary condition,When x =,0,t,xy,=,0,c,1,=,0,When x = l,/2,t,xy,= K,/2,c,2,= K,/,l,(f),(g),Sub.,(g),into,(a),(h),Integrate it,are the arbitrary function of y and x,and,Sub.,(f), (h),into,(d),Let,(i),Sub.,(i),into,(h), and consider,(f),Determine C,According to the condition of that the resulting force on the interface between over end and contacting area is equal zero, we can got that,When y = h/,2, we got that on the contacting interface.,外端影响系数,a,z,r,h,6.7 Quasi- static,axisymmetric,upsetting of circular cylinder,（,轴对称圆柱体镦粗）,Description of the process,1),Coulomb friction at the interface, and,m,is lower.,2),h,and,a,are the height and radius of the cylinder respectively.,3),Cylindrical coordinate.,r,dr,p,t,xy,=,m,p,s,r,s,r,+,d,s,r,Normal pressure distribution,Assumption,Cylindrical surface remain cylindrical.,Cylindrical surface,:,r.,Cylindrical surface,:,r +,dr,Two meridian planes,:,d,q,is the angle between the two meridian planes.,On the cylindrical surface and meridian planes the normal stresses are uniform, and there are no shear stresses.,1),Take an element.,s,r,s,r,+,d,s,r,2),Analysis of stresses exerted on the element.,d,q,dr,r,d,q,dr,r,d,q,/,2,m,p,s,r,s,r,+,d,s,r,s,q,s,q,r,r+dr,meridian surface,h,top,bottom,interface,p,and,m,p,Cylindrical surface stress area,3),Equilibrium equation.,S,r,= 0,s,r,r,d,q,h,s,r,s,r,+,d,s,r,(,r+dr,),d,q,h,s,r,+,d,s,r,s,q,dr,h,s,q,s,q,r,d,q,dr,p,p,m,p,Because,d,q,is very small, therefore,Dividing the equilibrium equation by,hd,q,and neglect the second order of the term, we can get the following equation,:,4),Yield criterion.,(A),Take 0z, or and,q,as the principal directions. From the incompressibility law we have,Differentiate the above equation, we got,Because,Therefore,However,From L-M equation,therefore,Axisymmetrical,stress state at any point of the cylinder,From v-M or Tr. Yield criterion,Y,is the yield stress in,uniaxial,stress state.,(,lode parameter,m,d,=,-1 ),Because,(B),(C),Sub,(B),and,(C),into,(A),(D),5),Determine A according to the prescribed stress boundary conditions.,At r = a.,From yield criterion.,Sub. Into,(D).,Therefore,6),Force exerted by the platens and mean platen pressure.,r,dr,Therefore,Expanding,into series,6.8,Bar and wire drawing through a conical die with and without back tension,Description of the process,Small semi-angle of conical die,a,Low friction coefficient,m,t,= m,p,At entry,r = R,1, tensile stress from back tension is,S,1,At exit,r = R,2, drawing stress is,S,2,a,R,1,S,1,o,R,2,S,2,Assumption,All points on a section normal to the drawing direction are in the same stress state, that is ,axisymmetrical,stress state.,Plane section normal to the drawing direction remain normal.,a,S,1,o,R,2,S,2,R,1,y,dr,r,dr,s,z,s,z,+,d,s,z,p,ds,r,m,p,(,At every point on a section,),s,z,is uniform,There is no shear stress on the section plane.,Stress analysis,Neglect the second and higher order of the term,S,z =,0,S,r =,0,Yield criterion.,(A),(B),Let,(C),Determine A according to the boundary condition,At entry,Therefore,Hence,At exit,Therefore,6.9 D,rawing and sinking of a thin-walled tube through a conical die,General consideration,Sinking of a thin-walled tube through a conical die,薄壁管减径拉拔,