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

Displacement and Strain

Displacement and Strain

Outline: Generalized DisplacementSmall Deformation Theory Continuum Motion & Deformation Strain & RotationPrincipal StrainsSpherical and Deviatoric StrainCylindrical Strain and Rotation: Spherical Strain and RotationStrain CompatibilityDomain Connectivity2

Outline • Generalized Displacement • Small Deformation Theory • Continuum Motion & Deformation • Strain & Rotation • Principal Strains • Spherical and Deviatoric Strain • Cylindrical Strain and Rotation • Spherical Strain and Rotation • Strain Compatibility • Domain Connectivity 2

Displacement.Conceptof displacement: coordinatedifference of the samematerial point intwo reference states? Displacement=Rigid-body translation+Rigid-body rotation+ Strain deformation: Rigid-body motion: the distance and angle among all materialpoints remain the same.Strain deformation: a material is said to be deformed or strainedwhen the distance or angle among material points is changed.We are not concerned with rigid-body motions in elasticity theory3

Displacement • Concept of displacement: coordinate difference of the same material point in two reference states. • Displacement = Rigid-body translation + Rigid-body rotation + Strain deformation • Rigid-body motion: the distance and angle among all material points remain the same. • Strain deformation: a material is said to be deformed or strained when the distance or angle among material points is changed. • We are not concerned with rigid-body motions in elasticity theory. 3

Small Deformation TheoryAdx=dx'-dx =u-uPtPYdxdx'oPPi(Undeformed)(Deformed)(Deformed)(Undeformed)Quouou. Taylor expansion of u w.r.t. uo:dzdxu=udy+Ozaxayu=u° +u.dx+..OvOvOvdzdx -d1V=Ozaxu, = u' +ui.,dx, +.ayowowowdzdx +dy-W=W△dx, = u, -u ~ ui,dx一ayOzax4

Small Deformation Theory , , d d d d o i i i j j o i i i i j j u u u x x u u u x                 o u u u x (Deformed) (Undeformed) d d d d d d d d d o o o u u u u u x y z x y z v v v v v x y z x y z w w w w w x y z x y z                               • Taylor expansion of u w.r.t. u o : dx dx d d d o      x x x u u  4

Small Deformation Theory: Displacement gradientOuOuouaxOz6avav)=6+0axOzOwowowayOxOz + Vu), strain tensor (symmetric)1=(uV-Vu), rotation tensor (anti-symmetric)2.Total displacementu, = u +(cu +)dx;?5

, , , , , 1 1 ( ) ( ) 2 2 i j i j j i i j j i ij ij u u u x y z v v v u u u u u x y z w w w x y z                                                  , , , , 1 1 ( ); , strain tensor (symmetric) 2 2 1 1 ( ); , rotation tensor (anti-symmetric) 2 2 ij i j j i ij i j j i u u u u                   u u u u Small Deformation Theory • Displacement gradient • Total displacement   d o i i ij ij j u u x        5

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

Sample ProblemDetermine the displacement gradient, strain and rotation tensors for the following displacementfield: u = Ax y , v = Byz , w = Cxz3, where A, B, and C are arbitrary constants. Also calculatethedualrotationvector(=(1/2)(Vxu)Ax?02AxyBz0Byuii=Cz303Cxz?[2AxyAx?/2Cz3 / 2Ax? / 2BzBy/23Cxz?Cz3 / 2By /20Ax? / 2-Cz3 /20-Ax? /2By /2Cz3 / 20-By / 2e2eiesBye, -Cz’'e, - Ax’e,a/ax0/oz0=(1/2)(V ×ualay2Cxz3AxyByz10

        2 , 3 2 2 3 2 , , 3 2 2 3 2 , , 3 1 2 3 2 2 0 0 0 3 2 / 2 / 2 1 / 2 / 2 2 / 2 / 2 3 0 / 2 / 2 1 / 2 0 / 2 2 / 2 / 2 0 1 1 2 / / / 2 i j ij i j j i ij i j j i Axy Ax u Bz By Cz Cxz Axy Ax Cz u u Ax Bz By Cz By Cxz Ax Cz u u Ax By Cz By x y z Ax y Byz Cxz                                         e e e ω u   3 2 1 2 3 3 1 2     By Cz Ax e e e Sample Problem 10 Determine the displacement gradient, strain and rotation tensors for the following displacement field: 2 3 u  Ax y , v  Byz , w  Cxz , where A, B, and C are arbitrary constants. Also calculate the dual rotation vector  = (1/2)(u)