Naslov
Full C1 continuity multiscale second-order computational homogenization approach

Autori
Lesičar, Tomislav ; Tonković, Zdenko ; Sorić, Jurica

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
11th World Congress on Computational Mechanics (WCCM XI) : abstracts / Onate, E. ; Oliver, X. ; Huerta, A. - Barcelona : Artes Gráficas Torres S.A., Huelva 9, 08940 Cornellà de Llobregat, Spain, 2014

ISBN
978-84-942844-7-2

Skup
World Congress on Computational Mechanics (11 ; 2014)

Mjesto i datum
Barcelona, Španjolska, 20.07.2014. - 25.07.2014

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Multiscale; Second-order computational homogenization; C1 continuity finite element; C1 continuity RVE; Generalized gradient periodic boundary conditions; Gradient displacement boundary conditions

Sažetak
In recent years, a special attention has been directed to investigate relations between the macroscopic properties of engineering materials and their microstructure. It is well known that the classical continuum mechanics cannot consider structural effects in the material at the microlevel and therefore, multiscale techniques for modeling on multiple levels using homogenization procedures have been developed. Several homogenization methods have been proposed, where the second-order computational homogenization scheme has been proven as most versatile tool, enabling consideration of the size effect. The multiscale algorithm comprising second-order homogenization procedure requires C1 continuity at the macrolevel and employs nonlocal continuum theory. The discretization of representative volume element (RVE) at the microlevel is usually performed preserving the standard C0 continuity and keeping classical boundary value problem with standard material models. However, due to the C1 - C0 transition, some inconveniences arise. Firstly, the microlevel second-order gradient cannot be related to the macrolevel as volume average, since such relation requires higher-order boundary conditions. Therefore, to prescribe full macrolevel second-order gradient, an alternative relation is considered, which leads to derivation of microfluctuation boundary integrals involving RVE boundary nodal displacements. As the second, the minor discrepancy exists in Hill-Mandel energy condition, since higher order variables are not included into microlevel computation, which requires a modified definition of the second-order stress as first moment of the Cauchy stress tensor. In this way, such approach does not satisfy volume average equivalence between the micro and macro second-order stress tensors. To overcome these shortcomings, the present contribution deals with a multiscale second-order computational homogenization algorithm keeping the C1 continuity at both the macrolevel and the microlevel under assumptions of small strains and linear elastic material behavior. Accordingly, the computational models at both levels are discretized by the C1 continuity three-node plane strain triangular finite element, developed and reformulated for multiscale analysis in the authors’ previous papers. To accelerate the solution process of the multi-scale problem, the reduced numerical integration scheme is used to evaluate the element matrices. Herein, a new scale transition methodology is derived in which the volume average of every macrolevel variable prescribed at the microlevel is explicitly satisfied without using boundary integrals to enforce equality of micro- and macro- variables. Furthermore, employing the Hill-Mandel energy condition the macrolevel stress tensor is extracted as volume average of microlevel stress tensor, without auxiliary definition of higher-order stress. The macroscopic consistent constitutive matrix is computed from the RVE global stiffness matrix using the standard procedures. A special attention is directed to the application of the gradient displacement- and generalized gradient periodic boundary conditions on the RVE. The algorithms derived are implemented into FE software ABAQUS via user subroutines. Finally, in order to test the performance of the proposed multiscale homogenization approach, several standard numerical examples are considered.

Izvorni jezik
Engleski

Znanstvena područja
Strojarstvo

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