(塑性加工模拟及自动控制)SimulationsMaterials2

1、( (塑性加工模拟及自动控制塑性加工模拟及自动控制)SimulationsMaterials2)SimulationsMaterials2Origin of Elastic deformation In case of elastic deformation, the volume changes. When a force (or pressure) is applied in one direction, there is a deformation in the lateral direction:beforeWith applied forceOrigin of Elastic def

2、ormation The lateral strain depend on the material. Its measure give the Poisson ratio noted (Greek letter nu) yy=-/ExxbeforeWith applied forceXYHooke law For elastic deformation, we have to use the Hooke law ij = Cijkl : kl , C is the stiffness tensor (刚度张量) C is a tensor of rank 4, but we write it

3、 as a matrix 6*6, which is for isotropic materials C = and = 2+ 0 0 0 2+ 0 0 0 2+ 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2xx yy zz xy xz yzGangduHooke law ij = ijkk+2ij ij = Kijkk + 2(ij 1/3*ijkk) with K: Bulk modulus (or incompressibility), =K-2/3* Lam parameter (lamda)=G: shear modulus (or rigidi

4、ty, mu), ij=1 if i=j, else 0 the Kronecker delta normal stress: xx = (xx+yy+zz)+2xx = K*V+2(xx-V/3) Shear stress: xy = 2xyHooke law ij = Sijkl : kl , S is the compliance tensor (柔度张量) ij=1/E*(1+)ij-ijkk E: Young modulus, = Poissons ratio (nu) S = 1/E*1 - - 0 0 0 - 1 - 0 0 0 - - 1 0 0 0 0 0 0 (1+) 0

5、0 0 0 0 0 (1+) 0 0 0 0 0 0 (1+) roudu zhangliangHooke law Relationships between K, , , E, : = E/2(1+) K = E/3(1-2) = +2/3* = E/(1-2)(1+) =K-2/3* E = 9K/(3K+) = (9+6)/(3-) = (3K-2)/2(3K+) = /2(-/3)Hooke law Uniaxial traction along X: yy = zz = - xx V= xx + yy + zz = xx*(1-2v)/E xx = (V)+2xx=EV/(1-2 )

6、(1+) + 2E/2(1+)*xx = xx /(1+) + E/(1+)*xx xx(1+-)/(1+) = E/(1+)xx xx = E xxApplications Compute the elastic deformation during a compression test of Mg alloy Compute the change of volume for Mg during a tensile test e = xx +yy +zz = -*zz/E -*zz/E + 1/E*zzApplications in uniaxial deformation:V= xx +yy +zz = xx*(1-2)/E Metal submitted to a pressure p= xx= yy= zz:V=p/E*(1-2)*3=p/K =0.5