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

Incompressible Spherically Symmetric Solids. Cauchy stress: 6=o, [rle,e, +oo[r]e,e, +0oe[rleseo, Op =O-: Deformation gradient:F= F,.[r, R]e,er + Fo,[r, R]e,ea + Fo[r,R]eeeo, Fo, = FaAALeft C-G deformation tensor:B=B,[r]e,e, +Boo[r]e,e, +Beo[rlesee, Bo = ]B00 Strain-displacement relations:drdrrFFFBB>00pD0RdRdRRIncompressibility:drr=1.J = det[F]dR(R)6

Incompressible Spherically Symmetric Solids • Cauchy stress:       , . rr r r r r r σ                  e e e e e e • Deformation gradient: F F r R F r R F r R F F     rr r R  , , , , . e e e e e e             • Left C-G deformation tensor:       , . B e e e e e e     B r B r B r B B rr r r         2 2 , , , rr rr dr r dr r F F F B B B dR R dR R                        • Strain-displacement relations:   2 det 1. dr r J dR R          F • Incompressibility: 6

Incompressible Spherically Symmetric Solids. Stress-strain relationauauauauauBB21p,3alal,al.alal,auauauauauB21O6p-0Oal,al,al2al,al,32doEquilibrium equations: +F=0-adr Traction BCs: O,[a]=Oa,O,m[b]=Ob. Displacement BCs: u, [a]= ua, u, [b]= up7

Incompressible Spherically Symmetric Solids • Stress-strain relation • Traction BCs: • Displacement BCs: 2 1 1 2 1 2 1 2 2 2 1 1 2 1 2 1 2 2 1 2 2 , 3 1 2 2 . 3 rr rr rr U U U U U I B I I B p I I I I I U U U U U I B I I B p I I I I I                                                                                     • Equilibrium equations:   2 0. rr rr r d F dr r             rr a rr b a b    , .   u a u u b u r a r b     , .   7

Pressurized Hollow Sphere: Incompressibilityqe3Pb: No body forces act on the sphere. Before deformation, the sphere hasinner radius A and outer radius BpThe solid is made from ane2incompressible Mooney-Rivlineisolid.Solution:Incompressibility implies:=3-α=R-A。=「4元r2dr=4元R2dRV =V.→r=(R-A+α')，R=(3-α3 +A3)8

• No body forces act on the sphere. • Before deformation, the sphere has inner radius A and outer radius B. • The solid is made from an incompressible Mooney-Rivlin solid. Pressurized Hollow Sphere: Incompressibility • Solution: • Incompressibility implies:     2 2 3 3 3 3 0 1 1 3 3 3 3 3 3 3 3 4 4 , r R a A V V r dr R dR r a R A r R A a R r a A                   8

Pressurized Hollow Sphere: Governing Equations: Deformation gradient and left C-G strain:drrFFBB= Bo三H1LRRdRRRStress-strain relationsI,=Bu = Bm +2Boo, 1,=(7-BxBh)==(Bm +2Be) -(B, +2B2)=(2B,m +Be)Beaauaui42U=(I,-3)+(I, -3)al,2′al,20m =2(u4 + μ, Bem)(Bm - Be0)+ p,(μu + μ,Boo)(Bm - Be)+ pGoo=000=0900Equilibrium equation:2R，R4R+)-(-等)dom +2(am-0m)+F,=0+C0,=μ一1dr9

• Deformation gradient and left C-G strain: Pressurized Hollow Sphere: Governing Equations 2 4 2 , , , rr rr dr r r r r F F F B B B dR R R R R                               • Stress-strain relations:                   2 2 2 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 , 2 2 2 2 2 3 3 , 2 , 3 2 2 2 2 1 . 3 rr rr kk rr ik ki rr rr rr rr I B B B I I B B B B B B B B B U U B B B p U I I I I B B B p                                                        • Equilibrium equation:   4 2 1 2 4 2 2 2 2 2 0 . rr rr r rr R R r R C r d F dr r r R r                              9

Pressurized Hollow Sphere: BCs· Boundary conditionsA42a2A4-p=μL2a4A[r=a,R= A:Om=-Pa -aB4r=b;R=B:Om=-Pb2B2bBPb=/山26462R2bB: Adding the two equations gives the expression for C121中Ll2Pa+ pbU+ 2B2α2042B4B2β22a2Although unnecessary, the Pressure p in the stress-strainrelations can be determined by comparing expressions ofthe radial stress10

• Boundary conditions Pressurized Hollow Sphere: BCs 4 2 1 2 1 2 4 2 4 2 4 2 1 2 1 2 4 2 4 2 2 2 2 1 1 2 ; : 2 2 ; : 2 2 2 1 1 2 2 2 a rr a rr b b A A A p C C r a R A p a a a r b R B p B B B p C C b b a b A b B                                                                                                            • Adding the two equations gives the expression for C. • Although unnecessary, the Pressure p in the stress-strain relations can be determined by comparing expressions of the radial stress. 10