东南大学：《固体力学基础》课程教学课件（英文讲稿）11 Simple Elastoplastic BVPs

Hollow Cylinder under Monotonic Pressure? In the plastic region a < r < c To simplify the calculation, we assume: dee = O, deP = 0 This assumption turns out to be exact for v = O.5 but isapproximate for other values of Poisson's ratio The plastic flow rule shows thatdep →o.=(a,+o)O=dep:g-+o。a.Yield criterion,-0。)°+(0。-0.) +(0.-0,))=/(0,-0。)aQ,=O。=20y-0,)=0y9。>0,9,<0=o。-r311

11 Hollow Cylinder under Monotonic Pressure • In the plastic region a < r < c • To simplify the calculation, we assume: • This assumption turns out to be exact for ν = 0.5 but is approximate for other values of Poisson’s ratio. • The plastic flow rule shows that 0, 0 e p z z d d         1 0 + 2 + 2 1 z p p z z r Y r d d                       • Yield criterion            1 3 2 2 2 2 2 4 3 2 0, 0 2 3 Y e r z z r r Y r r Y r                                         

Hollow Cylinder under Monotonic Pressure·In the plastic region a< r< c. Equation of static equilibrium20ylnr+Ddor+-+=-a :do.20,0UaV3rV3drdr? Integrate and apply the BCs at R = a20ylnr/a-o,[a]=-pa=.la-p320y20y20ylnr/a-d。-0rop.V3/3V3. Elastic strain:do, v(do。+dov(do, +do,dodae=0EEEE12

Hollow Cylinder under Monotonic Pressure • In the plastic region a < r < c • Equation of static equilibrium 12 • Integrate and apply the BCs at R = a r r r d F dr r       2    r 2 2 0 ln 3 3 r Y Y r d r D dr r             2 ln 3 2 2 2 , ln 3 3 3 Y r a r a Y Y Y r a a p r a p r a p                           , , 0. eee r z z r r z d d d d d d ddd E E E E                    • Elastic strain:

Hollow Cylinder under Monotonic Pressure. In the plastic region a < r < c. The plastic strains satisfy:dpdepds91QyOdpdde"(o,+0)-0.=0.(α, +0。+20.)(=9dop+deh+0。a: The elastic strains thus satisfy (plane strain condition):de+d=de+dedddo+d(ddo)EEE? Since the pressure is monotonically increasing, theincremental stress-strain relation can be integrated.)=5, +6 +)(1-2)(do, +do,) (+)(-2)(4du..2g+ur-1 d40ylnTas1a-2pEEV3dr13rdr(1+v)(1-2v)r20u.E313

Hollow Cylinder under Monotonic Pressure 13 • In the plastic region a < r < c • The plastic strains satisfy: • The elastic strains thus satisfy (plane strain condition): • Since the pressure is monotonically increasing, the incremental stress-strain relation can be integrated              d 4 2 1 1 1 2 ln 2 d 3 3 1 1 2 2 1 ln 3 1 2 r r Y Y r r r a Y r a u u d ru r a p d d r r r dr E r u r a p E C E r                                                       1 1 + , + 2 2 1 1 +2 0. 2 2 p p p p r r z z r Y Y p p p p r r r z r z Y Y d d d d d d d d                                                                                 2 1 1 2 e e r r z r r r d d d d d d d d d d d E E E                               

Hollow Cylinder under Monotonic Pressure? In the elastic region c < r < b2vp.c?b?(1+v)b2P.c2P.cT02b? _h2 -c2b?-cE(b? -c2? Form the radial stress in the plastic region, we obtain thepressure at the elastic-plastic boundary r = c20yInc/a20% lnc/a- pa=-P。 = P。= Pao,[c] -/313. The elastic-plastic boundary is located by noting that thestress in the elastic region must just reach yield at r = c.2% = 0.[] -0,[0] = Rc, 24 22pe20ylnnclaLb?-c? c?1-c2/b213(1-c2 /62)(1-c2 /b2)0, (1-c2 / b2)2 lnc/aPa20ylnc/a=p.O-V3V3V3V3Qy14

Hollow Cylinder under Monotonic Pressure 14 • In the elastic region c < r < b • The elastic-plastic boundary is located by noting that the stress in the elastic region must just reach yield at r = c. • Form the radial stress in the plastic region, we obtain the pressure at the elastic-plastic boundary r = c       2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 , 1 , ; 1 2 c c c c r z r p c p c p c b b b p c r u b c r b c r b c r E b c                                         2 2 ln ln 3 3 Y Y r a c c a c c a p p p p c a                       2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ln 3 3 1 1 2ln 1 1 2 , ln 3 3 3 3 Y Y c c r a Y a Y c a Y p c p b c c p c a b c c c b c b p c a c b c b p p c a                              

Hollow Cylinder under Monotonic Pressure: The constant of integration can be found by noting that theradial displacements in the elastic and plastic regimensmust be equal at r = c.(1+v)(1-2v) rC20y.1r. In the plastic region: u,DE21(1 +v) p.c2h. In the elastic region: u,E(b2-c2? Enforcing the displacement continuity condition2(1-v2)b2c22(1-v2)b2c220yInc/a.E(b? -c2)E(b? -c22(1-v2)b2c2 0y (1-c2/b2)2 (1-v2)c2V3EV3E(b?-c215

Hollow Cylinder under Monotonic Pressure • The constant of integration can be found by noting that the radial displacements in the elastic and plastic regimens must be equal at r = c. 15 • In the plastic region: • In the elastic region: • Enforcing the displacement continuity condition 1 1 2   2 ln 3 Y r a r u r a p E r        C              2 2 2 2 2 1 1 2 c r p c r b u E b c r                              2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 3 2 ln 3 1 3 Y a Y Y c p p c b c b c C E b c E b c b c c E b b c a c E                          