东南大学：《弹性力学》课程教学课件（英文讲稿）14 Energy Method and Variational Principle

Strain Energy Density for a General Stress State· Strain energy density of? Strain energy density oflinearly elastic material undernon-linearly elasticmaterial undergeneralized 3-D stress statesgeneralized 3-D stress10,ex+0,8, +0.8.states22+ty+tY+tyxydU。=o,daj·In Terms of Strain=odeode+ode28.+2GU.+tdyx+tydy+t.O(e +6,+8+G2222· In Terms of StressV1(1+V1+V1福v1木dyU.00Co1022E2E2EE1+v+a,+o.O2E2Fdx11

x y z dx dz dy 0 d d d d d d d d ij ij x x y y z z xy xy yz yz zx zx U                    • Strain energy density of non-linearly elastic material under generalized 3-D stress states • Strain energy density of linearly elastic material under generalized 3-D stress states 0 1 1 2 2 x x y y z z ij ij xy xy yz yz zx zx U                           11 Strain Energy Density for a General Stress State • In Terms of Strain • In Terms of Stress     0 2 2 2 2 2 2 2 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 ij ij kk ij ij ij kk jj ij ij x y z x y z xy yz zx U G G G                                             0 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 222 2 2 ij ij ij ij kk ij ij ij kk jj x y z xy yz zx x y z U E E E E E E                                                

Decomposition of Strain Energy DensityOO2a0.+pr17P木dydydyOXO.nd2d1dxdxdx(a) Spherical(b) Deviatoric stressU。=Uv+Uptensorstress tensor(1-2v3(1-2v). Volumetric energy density: Uy(o+o,+o.)12E6E: Distortion energy density:(1-2v(ar+a,+a)+o,+a, +2t,+2t +2t2)-1U,=U.-Uy2E6E1+v[1+}.+2t2+2t2+2t32E6E) +(g.-)-T+E6F12

(a) Spherical stress tensor (b) Deviatoric stress tensor = + x y z dx dz dy  y  x  z x y z dx dz dy  m  m  m x y z dx dz dy   y m    x m    z m  U U U 0 V D                         2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 xy yz z 0 x 2 2 2 2 2 1 2 6 1 1 222 2 6 1 1 1 222 2 2 6 D V x y z x y z xy yz zx x y z x y y x y z xy y z z x U U z zx x y z E E E E U E E E                                                                                    2 2 3 1 2 1 2 ( ) 2 6 UV m x y z E E         • Volumetric energy density:     • Distortion energy density: 12 Decomposition of Strain Energy Density

Strain Energy Density in terms of DisplacementSC..822(1222auduOwOv++oxax2ayazyQuavow1+G2OxOzd1av1 Quowow1十++22OzaxOzayEEv2(1 + v) (1 - 2v)2(1+ v)13

  0 , , 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 2 2 kk jj ij ij k k j j ij ij x y z x y z xy yz zx U G u u G G u v w u v u v w x y z y x G x y z u w v z x z                                                                                                                2 , 1 1 2 2 1 w y E E G                                  Strain Energy Density in terms of Displacement 13

Strain Energy Density for Plane Elasticity3-K3-KX3-K2(1 -KQu3-KvC1duOxax2 (1 - k)ayavavavauauOuLForplanestrain:x=3-4v:U.=GaxOxax2ay1-2voyoy3-vQuavauavauForplane stress:k=G2axoyoyax1 + vaxoy14

Strain Energy Density for Plane Elasticity                 0 2 2 2 2 2 2 2 2 1 1 3 3 2 2 2 2 1 2 1 3 2 2 1 1 3 2 2 1 x y xy x y U G G G u v u v u v G x y y x x y                                                                                                                        2 2 2 2 0 2 2 2 2 0 1 For plane strain: 3 4 : 2 1 2 3 1 For plane stress: : 1 2 1 u v u v u v U G x y y x x y u v u v u v U G x y y x x y                                                                                                                  14

Strain Energy Density for a General Stress State2c3P3Pxy,72c34cC11+v1 +vV2tUor2E2E2EE1 +vU = JJUdV=J"F Jdxdyd?2EE1+vdxdy1E2F9p2+dxdydxdyE2Hp?L?9PL(1+V)4Ec3Ec15

  2 3 2 2 2 2 2 2 0 1 2 2 0 0 0 2 2 0 2 2 2 6 0 3 3 , 1 , 0 2 4 1 1 1 2 2 2 2 1 1 d d d d 2 1 1 d d 2 1 9 1 9 d d 2 4 x xy y z yz zx x xy x x xy c L x xy c c L x xy c c L c P P y x y c c c U E E E E U U V x y z E E x y E E P x y x y E c E                                                                             2 2 2 2 2 0 2 2 2 3 1 d d 1 6 9 (1 ) 4 c L c P y x y c c P L P L Ec Ec               x y P L 2c Strain Energy Density for a General Stress State 15