东南大学：《弹性力学》课程教学课件（英文讲稿）03 Displacement and Strain

Continuum Rigid-body Motion & Deformation: Components of total displacement at a material point+ 8.,dx, + ,dxu.=uGeneralRigid-bodyStrainRigid-bodydisplacementdisplacementdisplacementrotation(Undeformed Element)(Rigid BodyRotation)(Vertical Extension)(ShearingDeformation)(Horizontal Extension)6

d d o i i ij j ij j u u x x      • Components of total displacement at a material point General displacement Rigid-body displacement Strain displacement Rigid-body rotation (Undeformed Element) (Rigid Body Rotation) (Horizontal Extension) (Vertical Extension) (Shearing Deformation) Continuum Rigid-body Motion & Deformation 6

Two-dimensional strain-displacement relationotB(x+dx,y):Ouu(x+dx, y)=u(x,jdxv(x,y+dy)axavv(x+dx, y)=v(x, ydxaxC(x, y+dy):auv(x,y)Oxu(x,y+dy)=u(x, y11dxayu(x+dx.yu(xy)yv(x,y+dy)=v(x,dyOnouOvauOu= dxdx:A'BdxdxdxXaxaxOxaxavOuOvOuaaudx +dxdxdyα ~tanatandidyaxaxaxOyayay

Two-dimensional strain-displacement relation                     d , : d , , d , d , , d ; , d : , d , d , , , . B x x y u u x x y u x y x x v v x x y v x y x x C x y y u u x y y u x y y y v v x y dy v x y dy y                       2 2 2 d d d d 1 2 u v u u A B x x x x x x x x                                   2 v x          1 d ; tan d d d , tan d d d . u x x v u u v v u x x x y y y x x y y x y                                              

Two-dimensional Geometric Deformation. Normal strainOudx-dx1ouA'B'- ABax8.=8.dxaxABOv:3-D Strain-displacementayOvA'C'- AC6,=8yACdyayrelationship.Engineering shear strainavOuow88ayOzaxOvOu元ZC'A'B'=α+β =Yxy(ouav2Oxa18x2ayaxShear strainavowS12Ozay1ou1Oyowou2axa11a2axOz8

1 d d , d 1 d d . d x xx y yy u x x A B AB u x AB x x v y y A C AC v y AC y y                                           • Normal strain ; 2 xy v u C A B x y                   • Engineering shear strain • Shear strain 1 1 . 2 2 xy xy v u x y                Two-dimensional Geometric Deformation , , 1 2 1 2 1 2 x y z xy yz zx u v w x y z u v y x v w z y w u x z                                                    • 3-D Strain-displacement relationship 8

Two-dimensional Rigid-body Rotationau: Rigid-body rotation around z-axisayOvOvOu1Ou02axOyOxOy. Integrate for constant rotationdyu=u°-o.yav[v* =v° +0.xaxdxx·3-D rigid-body rotation0,=-1/28k0jk=1/28kuk,j1OusOu2u=u°-のy+O,zwi = w3220x20x31=y°-0z+0_xOuiOu3W2=W132x3axi=w-0,x+o.y1Ou2aur3=w2120x10x29

Two-dimensional Rigid-body Rotation • Rigid-body rotation around z-axis 1 2 z v u v u x y x y                     • 3-D rigid-body rotation * * * o z y o x z o y x u u y z v v z x w w x y                     , 1 2 1 2 i ijk jk ijk k j        u 9 * * o z o z u u y v v x           • Integrate for constant rotation