Extension of the Nonuniform Transformation Field Analysis to linear viscoelastic composites in the presence of aging and swelling
Introduction
Constitutive relations of solid materials are usually formulated at the engineering, or macroscopic, scale. However, as the loadings become more complex, an accurate description of their response requires the introduction of more internal variables, whose physical meaning is not always clear and for which calibration of more material parameters is needed.
Micromechanical approaches provide an alternative to this phenomenological formulation of macroscopic constitutive relations, based on the observation that all solid materials are heterogeneous at a small enough scale. The present study develops a micromechanically based, reduced model for the effective behavior of linear viscoelastic composites with individual constituents which, in addition to being viscoelastic, undergo aging and swelling. This is typically the case of the nuclear fuel MOX (mixed oxides) under irradiation. Our objective is to predict the effective response of such composites under proportional and nonproportional mechanical loadings as well as under irradiation where aging and swelling play an important role.
Micromechanical models for linear viscoelastic composites fall, roughly speaking, into one of three categories:
- 1.
Analytical models. The earlier analytical models can be traced back to Hashin, 1965, Laws and Mc Laughlin, 1978 where, using, the Laplace transform, the problem is reduced to finding the effective properties of elastic composites with moduli depending on the Laplace parameter. It is now well-known that the effective properties of linear viscoelastic composites with short memory gives rise to long memory effects (Sanchez-Hubert and Sanchez-Palencia, 1978, Suquet, 1987, Barbero and Luciano, 1995). Even when the relaxation spectrum of the individual constituents is discrete (Dirac masses corresponding to a finite number of relaxation times), the effective spectrum of the composite may be a continuous functions with an infinite number of relaxation times (see Rougier et al., 1993, Beurthey and Zaoui, 2000, Masson et al., 2012 among others). Ricaud and Masson (2009) showed that for specific microstructures (two-phase composites whose overall elastic properties are given by one of the Hashin–Shtrikman bounds) the relaxation spectrum of the composites remains discrete. This implies (Ricaud and Masson, 2009, Vu et al., 2012) that the overall constitutive relations of such composites can be alternatively written with a finite number of internal variables. Conversely for composites with a continuous relaxation spectrum, an infinite number of internal variables is required and the advantage of an analytical model for subsequent use in a macroscopic numerical computation is lost. This has motivated the introduction of approximate models with a finite number of internal variables (or equivalently with a finite number of relaxation times) mostly based on the approximation of the continuous relaxation spectrum by Prony series (Rekik and Brenner, 2011, Vu et al., 2012).
The main advantage of the analytical models, exact or approximate, when they require only a finite number of internal variables, is that their use in a numerical simulation is not significantly higher than that of usual linear viscoelastic constitutive relations (obviously this cost depends on the number of internal variables, or relaxation times, in the model). A first limitation of the analytical models for viscoelastic composites is that they make use of the Laplace transform to convert a viscoelastic problem into an elastic one. This procedure does not apply rigorously to aging materials whose material properties depend on time. A second limitation of analytical models is that they only deliver the effective response of the composite and no information about the distribution of the fields (stress, strain) at the microscopic scale, except for the first moment (average) of the fields per phase. The same limitation applies to semi-analytical models based on the inclusion problem (Kowalczyk-Gajewska and Petryk, 2011). However, a more detailed information about the statistics of the fields in the phases (field distribution, intra-phase standard deviation) is often essential to predict the lifetime of structures which is governed by local values of the fields (damage or fracture). An approximation of the statistics of the fields in each phase up to second-order can be obtained by means of the effective internal variable theory of Lahellec and Suquet, 2007, Lahellec and Suquet, 2013. However this latter approach does not deliver effective constitutive relations and can only be used to obtain the response of the composite to a prescribed loading path.
- 2.
Full-field simulations. The response (overall and local) of a representative volume element of the composite can be simulated directly (among many other references, see for instance Michel et al., 1999, Zohdi and Wriggers, 2005, González and Llorca, 2007 for computational micromechanics in general). For linear viscoelastic constituents the unit-cell problem is either solved as an elastic problem after use of the Laplace transform (Yi et al., 1998) or directly by a standard time-integration scheme. The advantage of full-field methods is that they account for all details of the microstructure and provide an accurate (exact to round-off errors) description of the local fields. However, a first limitation of these methods is their computational cost. A second limitation is that full-field simulations provide a constitutive update in a step-by-step time integration, but do not provide a closed form expression for the constitutive relations. And even though the investigation of three-dimensional complex microstructures has become a common practice in the recent years, the coupling between computations at the microscopic scale (material) and at the macroscopic scale (structure) is still the exception, despite the attempts to perform computations at both scales simultaneously through nested Finite Element Methods (FEM2, see Feyel, 1999, Feyel and Chaboche, 2000).
- 3.
Reduced-order models. Reduced models aim at achieving a compromise between the two first class of approaches. On the one hand, they are based on numerical simulations at the microscopic scale (which can sometimes be replaced by analytical calculations) and on the other hand they deliver constitutive relations with a finite number of internal variables which can be used in macroscopic computations, at the expense of certain approximations which depend on the reduction method. One of the earliest method of this type is the Transformation Field Analysis (TFA) of Dvorak (1992) further developed in Dvorak et al. (1994) and extended to periodic composites by Fish et al. (1997). Assuming uniform eigenstrains within each individual constituent, Fish et al. (1997) derived an approximate scheme which they called, for a two-phase material, the two-point homogenization scheme. The original scheme and this extended scheme have been incorporated successfully in structural computations (Dvorak et al., 1994, Fish and Yu, 2002, Kattan and Voyiadjis, 1993). However, it has been noticed that the TFA induces a spurious kinematic hardening in the effective constitutive relations (Suquet, 1997) and that the application of the TFA to two-phase systems may require, in certain circumstances, a subdivision of each individual phase into several (and sometimes numerous) sub-domains to obtain a satisfactory description of the effective behavior of the composite (Michel et al., 2000, Chaboche et al., 2001, Michel and Suquet, 2003). The need for a finer subdivision of the phases stems from the intrinsic nonuniformity of the plastic strain field which can be highly heterogeneous even within a single material phase. As a consequence, the number of internal variables needed to achieve a reasonable accuracy in the effective constitutive relations, although finite, is prohibitively high. In order to reproduce accurately the actual effective behavior of the composite, it is important to capture correctly the heterogeneity of the plastic strain field.
This last observation has motivated the introduction in Michel et al., 2000, Michel and Suquet, 2003 of the Nonuniform Transformation Field Analysis (NTFA) where the (visco) plastic strain field within each phase is decomposed on a finite set of plastic modes which can present large deviations from uniformity. An approximate effective model for the composite can be derived from this decomposition where the internal variables are the components of the (visco) plastic strain field on the (visco) plastic modes. In addition, the NTFA provides localization rules which allow for the reconstruction of local fields upon post-processing of macroscopic quantities and it can predict local phenomena such as the distribution of the plastic dissipation at the microscopic scale (Michel and Suquet, 2009) under cyclic loading. A common feature shared by the TFA and the NTFA is that these methods are applicable in situations where the superposition principle applies, which in practice restrict their range of application to infinitesimal strains.
First applied to two-dimensional situations by Michel and Suquet (2004), the NTFA method has also been implemented in three-dimensional problems by Fritzen and Böhlke (2010) and extended to composites composed of a viscoelastic matrix containing elastic inclusions by Fritzen and Böhlke (2013). By contrast with this latter work, the three phases considered here are all viscoelastic. This opens a new choice for the definition of the modes, which can be defined in each individual constituents or over the whole volume element. Another distinctive feature of the present work is that swelling and aging of the phases are explicitly taken into account.
The paper is organized as follows. Section 2 presents the microstructure of MOX and the behavior of its individual constituents. The NTFA procedure is recalled in Section 3 and extended to account for swelling of the constituents. Two possible definitions of the modes, either defined in each individual phase (as done in Michel and Suquet, 2003) or on the entire volume element are introduced. The accuracy of both models is assessed by comparison with full-field simulations in Section 4 for non-aging materials and in Section 5 for aging materials.
Section snippets
Three-phase particulate composites
Mixed oxide fuel, commonly referred to as MOX fuel, will serve to illustrate the theory developed in this study. It is a nuclear fuel that contains more than one oxide of fissile material, usually consisting of plutonium blended with natural uranium, reprocessed uranium, or depleted uranium (Oudinet et al., 2008). It is therefore of composite with, roughly speaking, three distinct phases. A brief account on its microstructure and on the behavior of its individual constituents is given here for
Nonuniform transformation fields
The basic feature of the NTFA theory (Michel and Suquet, 2003, Michel and Suquet, 2004) is a decomposition of the viscous strain on a set of a few, well-chosen, fields, called modes:where
- •
the modes are incompressible tensorial fields (tr ()=0). The choice of these modes is essential in the method and will be discussed in Section 3.4,
- •
the ’s are the generalized viscous strains associated with each mode for which evolution equations will
Non-aging constituents
To check the accuracy of the two NTFA approaches, with global and local modes respectively, four tests have been performed. In this section, aging of the constituents is not taken into account.
- Test 1.
Radial monotonic loading along a stress direction which has not been used for the identification of the modes, involving tension and torsion along the first axis:
The volume element is loaded at constant strain-rate for 10 s.
- Test 2.
A creep test in uniaxial tension in
Composites with aging constituents
When the constituents of the composite are subject to aging, care should be used in the choice of the modes on which the decomposition (6) is based. By construction the modes do not depend on time, unlike the material properties of the constituents which, because of aging, vary with time. Therefore it is not clear which material data should be used in the tests from which the modes are constructed. A first possibility is to construct the modes as in Section 4 with the initial material
General conclusion
The present study has shown the capability of the Non-uniform Transformation Field Analysis to predict the overall response, as well as the distribution of the local fields, in three-phase viscoelastic composites with three-dimensional microstructure. In particular, its ability to handle individual constituents undergoing aging and swelling, in addition to creep, has been demonstrated. This situation is typically encountered in the nuclear fuel MOX under irradiation which has been chosen to
Acknowledments
J.C. Michel and P. Suquet acknowledge the financial support of the French Commissariat à l’Énergie Atomique via Grant CEA-EDF-CNRS 038279, of the Labex MEC and of A∗Midex through Grants ANR-11-LABX-0092 and ANR-11-IDEX-0001-02. R. Largenton acknowledges the financial support of Électricité de France via the COMCRAY project. The authors are indebted to R. Masson and G. Thouvenin for fruitful discussions.
References (44)
- et al.
Micromechanical formulas for the relaxation tensor of linear viscoelastic composites with transversely isotropic fibers
Int. J. Solids Struct.
(1995) - et al.
Structural morphology and relaxation spectra of viscoelastic heterogeneous materials
Eur. J. Mech. A/Solids
(2000) - et al.
Towards a micromechanics based inelastic and damage modeling of composites
Int. J. Plast.
(2001) - et al.
An overview of the proper generalized decomposition with applications in computational rheology
J. Non-Newtonian Fluid Mech.
(2011) Multiscale FE2 elastoviscoplastic analysis of composite structures
Comput. Mater. Sci.
(1999)- et al.
FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials
Comput. Methods Appl. Mech. Eng.
(2000) - et al.
Computational plasticity for composite structures based on mathematical homogenization: Theory and practice
Comput. Methods Appl. Mech. Eng.
(1997) - et al.
Computational mechanics of fatigue and life predictions for composite materials and structures
Comput. Methods Appl. Mech. Eng.
(2002) - et al.
Reduced basis homogenization of viscoelastic composites
Compos. Sci. Technol.
(2013) - et al.
Virtual fracture testing of composites: a computational micromechanics approach
Eng. Fract. Mech.
(2007)