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Multiscale Modeling of Heterogeneous Materials Using Second-Order Homogenization

New demands on reliability and safety, together with the applications of new materials and new production technologies, can only be realized by advanced structural analysis methods involving realistic description of material behavior with microstructural effects. Classical continuum mechanics assumes material homogeneity, therefore, for comprehensive assessment of structural integrity and reliability, an analysis at the microlevel is unavoidable. In this framework, whole new branch of numerical methods arise, concerned with multiscale modelling of material behaviour using homogenization procedures. Basically, this computational approach is based on the solution of two boundary value problems, one at each length scale. The results obtained by the simulation of a statistically representative sample of material, named Representative Volume Element (RVE), are used as input data at the macrolevel. Based on the micro-macro variable dependence, first- and second-order homogenization procedures are available. The multiscale analysis using the first-order computational homogenization scheme allows explicit modeling of the microstructure, but retains the essential assumptions of continuum mechanics. It is based on the principles of a local continuum and microstructural size is irrelevant. Recently developed second-order homogenization framework represents extension of the first-order homogenization from the mathematical aspect. The formulation relies on a nonlocal continuum theory with microstructural size as an influential parameter. Accordingly, in the finite element setting, C1 continuity is required at the macrolevel. An important problem in the second-order homogenization framework is the scale transition methodology due to C1 - C0 transition at the microlevel. Higher-order gradients at the macroscale cannot be defined on the RVE as volume averages. Also, transfer of the full second-order gradient tensor from macro- to the microlevel is not possible without additional integral relation. On the other hand, higher-order stress at the coarse scale cannot be explicitly averaged, since no higher-order boundary value problem is defined at the microlevel. In this research, a new multiscale algorithm using second-order computational homogenization is developed, where previously mentioned issues are circumvented by introduction of higher-order continuum at the microlevel. At first, classical C1 - C0 algorithm has been established, firstly for small strains, afterwards for a large strain case. In this framework, a distinct approach has been used at the macrolevel, which is in this research discretized by the fully displacement based C1 finite elements, contrary to the usually employed C0 finite elements based on the mixed formulation. Finite element formulation has been re-established for application in multiscale framework. Also, series of patch tests have been conducted for verification of the element. The element, as complete multiscale setting has been implemented into commercial finite element software ABAQUS through user subroutines written in FORTRAN programming language and PYTHON scripts. Implementation aspects regarding microfluctuation integral which arises due to continuity degradation have been examined. Several numerical integration techniques have been tested with emphasis on physically realistic RVE behaviour. In the small strain case material nonlinearity has been considered, which is extended to the geometrical nonlinearity. Having defined multiscale algorithm and second-order computational homogenization scheme, a new multiscale approach has been developed, preserving C1 continuity at the microlevel. Linear elastic material behavior and small strains have been adopted, where microlevel is described by the Aifantis strain gradient elasticity theory. In this case, both levels are discretized by the same C1 finite element. At the macrolevel generalized Aifantis continuum theory is established accounting for heterogeneities, since all the relevant information come from the RVE. Scale transition methodology has been derived, where every macrolevel variable is derived as true volume average of its conjugate on microscale. Displacement gradients at the coarse scale are imposed on the RVE boundaries through gradient- displacement and generalized periodic boundary conditions. By the virtue of higher-order continuum adopted at the microlevel, displacements as displacement derivatives are prescribed or related by periodicity equations. Besides, in Aifantis theory microstructural parameter l2 appears, as a measure of nonlocality. So, in the new C1 multiscale setting next to RVE size, another intrinsic nonlocality parameter is available. In the end, efficiency of the derived algorithms has been demonstrated by number of illustrative examples.

heterogeneous material; multiscale; C1 homogenization; C1 finite element; RVE; gradient boundary conditions

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