Finite-volume enabled transformation field analysis of periodic materials
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Finite-volume enabled transformation field analysis of periodic materials Marcio A. A. Cavalcante • Marek-Jerzy Pindera
Received: 3 December 2012 / Accepted: 12 February 2013 Ó Springer Science+Business Media Dordrecht 2013
Abstract The transformation field analysis (TFA) proposed by Dvorak et al. in a sequence of papers in the 1990s is an important conceptual cornerstone of the elastic–plastic analysis of heterogeneous materials. However, the need for highly discretized unit cells required to attain converged homogenized response using finite-element based calculation of the plastic influence matrices employed in TFA simulations has given rise to further developments, including the recent nonlinear TFA approach. This variant leverages characteristic plastic modes that arise in elastic–plastic heterogeneous materials. Herein, we re-visit the TFA approach in the context of periodic materials with large phase moduli contrast, and first quantify the unit cell discretization required to attain the same level of convergence as with full unit cell finite-element based analysis. Subsequently we demonstrate that the finitevolume based calculation of strain concentration and plastic influence matrices requires substantially smaller unit cell discretizations to achieve the same degree of macroscopic and microscopic level accuracy, resulting in large execution time reductions and fewer parameters that describe the underpinning plastic deformation mechanisms. Further reductions may be achieved by explicitly leveraging plastic field
M. A. A. Cavalcante M.-J. Pindera (&) Civil & Environmental Engineering Department, University of Viginia, Charlottesville, VA 22904-4742, USA e-mail: [email protected]
localization that assumes distinct spatial distributions or characteristic modes. Keywords Transformation field analysis Micromechanics Homogenization Finite-volume theory Finite-element analysis
1 Introduction Efficient determination of the homogenized elastic– plastic response of heterogeneous materials that could be integrated into a multiscale analysis algorithm continues to be a challenging undertaking given the complex microstructure-induced stress and plastic strain fields. Accurate determination of both homogenized response and local fields requires detailed solution of a unit cell or a representative volume element problem, depending on whether statistically homogeneous or periodic material microstructures are considered (Pindera et al. 2009). This is typically accomplished using numerical or semi-analytical techniques based on a sufficiently detailed discretization of the investigated domain representative of the heterogeneous material-at-large. Thus the choice of techniques capable of capturing field variable distribution details with sufficient fidelity is limited to variational-based, transform or finite-volume techniques. Hence the problem is intrinsically computationally intensive which is compounded by the implicit dependence of local displacement fields on plastic strains.
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M. A. A. Cavalcante, M
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