an orthotropic continuum model for the analysis of masonry structures

TNO-95-NM-R0712 1995 9whereúκ t = úε p α = ⎛ 1 /2⎝ 1 / 2 (úεúε p ) T Q úεúε p ⎞+⎠1 / 2 π T úεúε p , (20)⎡ 1/⎢ 2⎢ − 1 / 2Q = ⎢⎢ 0⎢ 0⎣− 1 / 21/ 20000000⎤⎥⎥⎥⎥⎥⎦0. (21)012α g2.1 Return mapping algorithm - Tension regimeThe integration of the constitutive equations given above is a problem of evolution that can beregarded as follows. At a stage n the total strain and plastic strain Þelds as well as the hardeningparameter (or equivalent plastic strain) are known:{ε n , ε p n, κ t, n } given data . (22)Note that the elastic strain and stress Þelds are regarded as dependent variables which can bealways be obtained from the basic variables through the relationsε e n = ε n − ε p n and σ n = D ε e n . (23)Therefore, the stress Þeld at a stage n+1is computed once the strain Þeld is known. The problemis strain driven in the sense that the total strain ε n+1 is trivially updated according to the exact formulaε n+1 = ε n +∆ε n+1 . (24)It remains to update the plastic strains and the hardening parameter. These quantities are determinedby integration of the ßow rule and hardening law over the step n → n+1. In the frame of afully implicit Euler backward integration algorithm this problem is transformed into a constrainedoptimization problem governed by discrete Kuhn-Tucker conditions as shown by Simo et al.(1988). It has been shown in different studies, e.g. Ortiz and Popov (1985) and Simo and Taylor(1986), that the implicit Euler backward algorithm is unconditionally stable and accurate. Thisalgorithm results in the following discrete set of equations:ε n+1 = ε n +∆ε n+1σ n+1 = σ trial −∆λ t, n+1 D ∂g ⎪1⎪∂σ⎪ n+1, (25)ε p n+1 = ε n p ∂g 1⎪+∆λ t, n+1 ⎪∂σ⎪ n+1κ t, n+1 = κ t, n +∆κ t, n+1in which ∆κ t, n+1 results from the integration of the rate equation, eq. (19), and the elastic predictorstep returns the value of the elastic trial stress σ trial asσ trial = σ n + D ∆ε n+1 . (26)The above equations must be satisÞed and simultaneously the yield criterion must be fulÞlled