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