Scale transitions in solid mechanics based on computational ... - Cism

multi-phase material, if the analysis is performed on a unit cell <strong>in</strong>stead of an RVE.As the simplest unit cell, a piece (for example a square or cube) of the matrix materialconta<strong>in</strong><strong>in</strong>g a s<strong>in</strong>gle heterogeneity (e.g. <strong>in</strong>clusion or void) could be suggested. The useof such a unit cell implicitly assumes a regular arrangement of the heterogeneities <strong>in</strong> thematrix, which contradicts the observations that almost all materials have a non-periodicor even spatially random microstructural composition. Examples are precipitates <strong>in</strong> metalalloys arranged randomly by their nature and artificial fiber re<strong>in</strong>forced composites, possess<strong>in</strong>ga non-regular distribution of the fibers due to the production process. At thesame time, several experimental evidences exist show<strong>in</strong>g that the spatial variability <strong>in</strong> themicrostructure significantly <strong>in</strong>fluences the overall behaviour and particularly the fracturecharacteristics of composites, as reported <strong>in</strong> [104, 105].Different authors, e.g. [46, 47, 57, 70], have performed a comparison of the overall compositeresponses result<strong>in</strong>g from the modell<strong>in</strong>g of regular and random structures. Theyhave found a significant response difference <strong>in</strong> the plastic regime, while there is almost nodeviation <strong>in</strong> elastic regime. Also it has been shown [106], that soften<strong>in</strong>g behaviour of aregularly composed structure may change to harden<strong>in</strong>g <strong>in</strong> the case of a random composition.Most of these considerations, except for the latter, have been performed for smalldeformations, very simple elasto-plastic behaviour and relatively stiff <strong>in</strong>clusions (fibers).In this section the overall behaviour of regular and random structures is compared at largedeformations, non-l<strong>in</strong>ear history dependent material behaviour, for voided material (anappropriate approximation for material with soft <strong>in</strong>clusions). Apart from the calculationson the microstructural cell (tensile configuration), also a full multi-scale analysis (purebend<strong>in</strong>g) of both regular and random structures is presented.A material with a 12% volume fraction of voids is considered. The regularly stackedstructure is modelled by a square unit cell conta<strong>in</strong><strong>in</strong>g a s<strong>in</strong>gle hole (figure 11a). For themodell<strong>in</strong>g of a random structure 10 different unit cells with non-regular arrangements ofvoids with a distribution of void sizes have been generated (figure 11b). The averagedbehaviour of these 10 unit cells is expected to be representative for the real randomstructure with a given volume fraction of heterogeneities. Us<strong>in</strong>g several small non-regularunit cells <strong>in</strong>stead of one larger RVE also allows to estimate the amount of deviation ofthe apparent properties obta<strong>in</strong>ed by the unit cell modell<strong>in</strong>g, from the effective values fordifferent types of material models and load<strong>in</strong>g histories.In the subsequent sections a comparison is performed for three different constitutive modelsof the matrix material: hyper-elastic, elasto-visco-plastic with harden<strong>in</strong>g and elastovisco-plasticwith <strong>in</strong>tr<strong>in</strong>sic soften<strong>in</strong>g. First uniaxial extension (under plane stra<strong>in</strong> conditions)of a macroscopic sample is considered. Because <strong>in</strong> this case the macroscopicdeformation field is homogeneous a full micro-macro modell<strong>in</strong>g is not necessary and ananalysis of an isolated unit cell with adequate boundary conditions (periodic) suffices.In the last section the results of a micro-macro simulation of bend<strong>in</strong>g us<strong>in</strong>g random andregular microstructures are compared.Elastic behaviour, tensionFirst, a comparison of the overall behaviour of regular and random structures is carriedout for the case of hyper-elastic behaviour of the matrix material, modelled as a compressibleNeo-Hookean material. The material parameters used <strong>in</strong> the calculations are30