“Computational Civil Engineering - "Intersections" International Journal

“Computational Civil Engineering 2005”, International Symposium 187Obviously, Eq. (32) holds at all times during loading. Using an incrementalexpression, we havewhere fEq. (33) and we obtain∫ Ω∫ Ω∆ { } n is the change of { } nT nn[ ] ∆{ } dΩ + ∆{ f } = 0B σ (33)f during ∆ tn= tn+1 − tn. Substitute Eq. (24) intoT−[ ] ⎜⎛n nα ~h nB [ D] ( ∆{} ε − { ε } ∆t) − A ⋅α⋅ t ⋅ [ D] ⋅ [ R] ⋅ { σ } ⎟⎞dΩ + ∆( f )⎝~ 1 n⋅ &vp n= 0 (34)⎠Replacewhere∆ {} εnby[ B] ∆ { d} n and Eq. (34) givesn n[ K ] ∆{ d} − ∆{ Ψ} = 0T(35)~= dΩnT n[ K ] [ B] [ D] [ B]T∫ Ω(36)∆T n n− T n h n{ Ψ} =∫ [ B] [ D] { ε vp} ∆tndΩ + A ⋅α⋅ t ⋅∫[ B] [ D] [ R]{ σ } dΩ − ∆( f )andΩ∆{} dn~ α 1 ~n& (37)Ωis the increment of total deformation at the nth time step.Initial conditions of visco-elastoplastic computation are the equilibrium ofelasticity. After the nth time step, displacement and stress increments are evaluatedby Eqs. (35) and (30). Substitute them into Eq. (25) and the increment ofviscoplastic strains can be determined. Stresses and viscoplastic strains calculatedby Eqs. (30) and (31) are then treated as the initial conditions during the nextiteration. Computation is repeated in the same way until the whole loading time iscovered.It should be pointed out here that convergence of computation is not involved inthe stated procedure since viscoplasticity goes on as long as the loading is applied.As stated in the foregoing sections, such phenomena were observed in theexperiments and are reflected by the viscous flow function Eq. (19).5. CHOICE OF PARAMETERS FOR COMPUTATIONSFrom the constitutive equations of multidimensions, it is easy to see that elastic andviscoelastic parameters determined from tests are utilized directly in the