成型力学课件3Strain Analysis教学教材.pptVIP

材料成形力学（双语教学） 3 Strain Analysis Deformation Process Pure deformation Rigid body displacement Rotation(旋转) Translation（平移） The analysis of strain is not concerned with rigid body displacement is concerned with relative displacement Relative displacement is the displacement of point in a body relative to one another when the body deformed. 3.1 Infinitesimal deformation （无限小变形） dx dz dy z y x O A G F E D C B H (x,y,z) (x+dx ,y+dy, z+dz) A G F E D C B H A’ G’ F’ E’ D’ C’ B’ u w v (x+ u, y+ v ,z + w) (x+dx)+(u+du) (y+dy)+(v+dv) (z+dz)+(w+dw) 3.1.1 infinitesimal strain at a point 1) Relative displacement tensor A G F E D C B H A’ G’ F’ E’ D’ C’ B’ u w v (x+ u, y+ v ,z + w) (x+dx)+(u+du) (y+dy)+(v+dv) (z+dz)+(w+dw) The disp. of A: u v w (ox) (oy) (oz) The disp. of B: Rigid body disp. is the same disp.as that of A. Relative disp.: (A B) du , dv, dw --- produce strain (Infinitesimal quantities) uB = u+du vB = v+dv wB = w+dw u,v,w are the continuous function of coordiates x,y,z,: u = u(x,y,z), v = v(x,y,z) w = w(x,y,z) A(x,y,z) disp. uA=u(x, y ,z), vA=v(x ,y ,z) wA=w(x, y, z) B(x+dx,y+dy,z+dz) disp. uB=u(x+dx,y+dy,z+dz), vB=v(x+dx,y+dy,z+dz), wB=w(x+dx,y+dy,z+dz) Infinitesimal element and continuous function uB=u(x+dx,y+dy,z+dz), vB=v(x+dx,y+dy,z+dz), wB=w(x+dx,y+dy,z+dz) can be expanded into the Taylor?s series. uB = u+du = u(x+dx,y+dy,z+dz), +[terms of higher powers of dx, dy and dz] Since u=u(x,y,z) is assumed to be a very small quantity, these latter terms can be neglected. Then Similarly 2) The meaning of the components of the relative displacement tensor Project（投影）the parallelepiped before and after deformation into the xoz, xoy and yoz planes Take the xoz plane as an example x z O z w x u A C F E A’ C’ E’ F’ dz dx A C F E A’ C’ E’ F’