100 Riccardo Rossi,Vitaliani Renato,and Eugenio Onate This is achieved by a two step procedure,based on a phase of assessment of the state of the membrane and on a phase of modification of the material tangent matrix. Many different choices are theoretically possible in combining the two dif- ferent phases,however in the writer experience,iterative application of the wrinkling correction inside the same time step leads generally to a very slow or unstable convergence behavior.The proposed solution procedure is therefore based on a“explicit'”approach in the form standard pseudo-static solution step check state of each element ·modify material ·go to next“time”step This procedure is very efficient as it takes full advantage of the pseudo-static solution procedure the only additional cost being linked to the evaluation of the state and to the penalization of the constitutive matrix.As during each time step the material is "constant",no additional source of non-linearity is introduced therefore the element retains its convergence properties.The stabilization of the stress-field is guaranteed by the dynamic process that, together with the stabilization introduced in the material model effectively damps out the oscillations. The reader should note that the aim of the proposed technique is to get a reliable static solution.There is absolutely no guarantee that "on the way" to the static solution the wrinkling procedure converges inside each time step, however,when all the movement is dissipated so the structure reached the final configuration,wrinkling arrived to a constant solution. Assessment of the state of the membrane One of the crucial steps in the procedure is the evaluation of the state of the membrane.In particular it is necessary to "decide"if the membrane is (or rather should be)in biaxial tension,in uniaxial tension or completely unstressed because of the formation of wrinkles.The assessment procedure, is based on the introduction of the fictitious stress o*that represents the stress that would exist on the membrane if formation of the wrinkles was not allowed.This is related to the total stress from the relation [o]=Doriginal][E} (59) the principal direction of o*can be calculated as c1=01+022;c2=011-022;c3= 号)2+(o)2 of=cI+c2 air c1-c2 a'tan- (60)100 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate This is achieved by a two step procedure, based on a phase of assessment of the state of the membrane and on a phase of modification of the material tangent matrix. Many different choices are theoretically possible in combining the two dif￾ferent phases, however in the writer experience, iterative application of the wrinkling correction inside the same time step leads generally to a very slow or unstable convergence behavior. The proposed solution procedure is therefore based on a “explicit” approach in the form • standard pseudo–static solution step • check state of each element • modify material • go to next “time” step This procedure is very efficient as it takes full advantage of the pseudo–static solution procedure the only additional cost being linked to the evaluation of the state and to the penalization of the constitutive matrix. As during each time step the material is ”constant”, no additional source of non-linearity is introduced therefore the element retains its convergence properties. The stabilization of the stress-field is guaranteed by the dynamic process that, together with the stabilization introduced in the material model effectively damps out the oscillations. The reader should note that the aim of the proposed technique is to get a reliable static solution. There is absolutely no guarantee that “on the way” to the static solution the wrinkling procedure converges inside each time step, however, when all the movement is dissipated so the structure reached the final configuration, wrinkling arrived to a constant solution. Assessment of the state of the membrane One of the crucial steps in the procedure is the evaluation of the state of the membrane. In particular it is necessary to “decide” if the membrane is (or rather should be) in biaxial tension, in uniaxial tension or completely unstressed because of the formation of wrinkles. The assessment procedure, is based on the introduction of the fictitious stress σ∗ that represents the stress that would exist on the membrane if formation of the wrinkles was not allowed. This is related to the total stress from the relation [σ∗] = [Doriginal] : {E} (59) the principal direction of σ∗ can be calculated as c1 = σ∗ 11 + σ∗ 22 ; c2 = σ∗ 11 − σ∗ 22 ; c3 = 3 ( c2 2 )2 + (σ∗ 12)2 σ∗ I = c1 + c2 ; σ∗ II = c1 − c2 ; α∗ = tan−1 2σ∗ 12 c2  ; (60)