TAILIEUCHUNG - Computational Plasticity- P7

Computational Plasticity- P7: Despite the apparent activity in the field, the ever increasing rate of development of new engineering materials required to meet advanced technological needs poses fresh challenges in the field of constitutive modelling. | On Multiscale Analysis of Heterogeneous Materials 175 Substituting 31 into 32 the overall tangent modulus representation is obtained as 33 __ 1 B T l V Db l K lin Clearly the modulus Cl is given as a function of the boundary coordinate matrix Dbjl defined in 29 the condensed stiffness matrix KBn and the global coordinate matrix Djlobal l outlined in 28 . Finally we remark that using 33 the tangent moduli can be computed for heterogeneous material with arbitrary microstructures. When using this tangent modulus the quadratic rate of convergence is attained at the macroscopic level. Periodic Displacements and Antiperiodic Traction on the Boundary In order to discretise the continuum model of the periodic boundary conditions described in the nodes of the mesh are partitioned in four groups 1 ni interior nodes 2 np positive boundary nodes which are located at the top and right of the microstructure boundary dV of the RVE 3 np negative boundary nodes which are located at the bottom and left of the microstructure boundary dV of the RVE and 4 nc 4 node at the corners. More details on these discrete constraints are given in 1 . Partitioning of Algebraic Equations The partition of the nodal displacements and internal forces for the periodic boundary condition is as follows Li u Lp u Ln u Lc u and f f i fp f n fc Li f Lp f Ln f Lc f 34 Here Li Lp Ln and Lc are the connectivity matrices which define respectively the interior contribution the contribution of positive boundary nodes the one from their corresponding negative boundary nodes and finally the contribution from the nodes at the corners. In correspondence to 34 the tangent stiffness matrix is partitioned in the following way K_ dfint du kii kip kin kic kpi k kpp k pn k pc kni k np k nn k nc kci k cp k cn k cc Li K LiT Li K LpT Li K LnT Li K LcT Lp K LT Lp K LT Lp K LT Lp K LT Ln K LT Ln K LJ Ln K LT Ln K LT Lc K LiT Lc K LpT Lc K LnT Lc K LcT 35 176 D. PeriC . de Souza Neto . Carneiro Molina and

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