东南大学：《固体力学基础》课程教学课件（英文讲稿）10 Simple Hyperelastic BVPs

Simple Hyperelastic BVPsmi@se.ed.cn

Simple Hyperelastic BVPs

Outline·Theoryof hyperelasticity（超弹理论回顾）·Incompressible spherically symmetric solids（不可压缩中心对称体）Pressurizedhollow sphere（压力球腔）>Governing equations（控制方程）》Boundarycondition（边界条件）>Displacementvs.Pressure（位移与压力函数关系）>Radial stress distribution（径向应力分布）>Hoopstressdistribution（箍筋应力/周向应力）2

Outline • Theory of hyperelasticity（超弹理论回顾） • Incompressible spherically symmetric solids（不可压缩 中心对称体） • Pressurized hollow sphere（压力球腔） 2  Governing equations（控制方程）  Boundary condition（边界条件）  Displacement vs. Pressure（位移与压力函数关系）  Radial stress distribution（径向应力分布）  Hoop stress distribution（箍筋应力/周向应力）

Summary of the Theory of HyperelasticityThe solid is stress free in its undeformed configurationTemperature changes during deformation are neglectedThe solid is incompressible.Re2RelDeformedOriginale3ConfigurationConfiguration3

• The solid is stress free in its undeformed configuration. • Temperature changes during deformation are neglected. • The solid is incompressible. Summary of the Theory of Hyperelasticity 3

Summary of the Theory of Hyperelasticity. Strain-displacement relations: B, = FiFjk, F, =O, +ujj Incompressibility: J = det[F] = 1BB.I, = BkkStress-strain relationdauauauauau1B. BB21nimimalal,3alal,alz00u+F, = 0.Equilibrium equationsayiTraction BCs on S: ,n, =tj Displacement BCs on Su: u, = u,4

Summary of the Theory of Hyperelasticity , , B F F F u ij ik jk ij ij i j     J   det 1 F • Strain-displacement relations: • Incompressibility: • Stress-strain relation • Traction BCs on St : • Displacement BCs on Su :   2 1 2 1 1 1 2 1 2 1 2 2 1 , , 2 2 2 . 3 kk ik ki ij ij ij im jm ij I B I I B B U U U U U I B I I B B p I I I I I                                             • Equilibrium equations: 0. ij j i F y     ij i j  n t  i i u u  4

Incompressible Spherically Symmetric Solidse3. Coordinates in undeformedeRconfiguration { R, Φ, @} Coordinates in deformedveoRconfiguration (r, P,0e20eiPoints only move radially, dueto spherical symmetryr= f(R)0=00= d.Position vectorin the undeformed solid:x=Re,Position vector in the deformed solid:y= re,=f(R)e. Displacement vector: u = y - x = re, - Re, = (f(R) - R)e5

• Coordinates in undeformed configuration Incompressible Spherically Symmetric Solids     • Coordinates in deformed configuration R, ,   r, ,   • Points only move radially, due to spherical symmetry 5