Renald Brenner

In situ diffraction strain analysis of elastically deformed polycrystalline thin films, and micromechanical interpretation

In situ tensile tests have been carried out under synchrotron radiation on supported gold (Au) thin films exhibiting a pronounced crystallographic texture. The 2θ shift of X-ray diffraction lines has been recorded for different specimen orientations and several loading levels in the elastic domain. The data obtained demonstrate the large strain heterogeneities generated within the specimen because of the intergranular interactions associated with the large elastic anisotropy of Au grains. To interpret these results, the use of a multi-scale micromechanical approach is unavoidable. The theoretical background of such methods is described, and the points where exact results can be obtained and where approximations have to be introduced are highlighted. It is shown that the Vook–Witt model, for which a general formulation is provided, is the exact solution for polycrystals exhibiting a laminate microstructure, which is a significant departure from the standard thin-film microstructures. Among several standard models used in the field, the self-consistent model is the only one that reproduces the experimental data correctly. This is achieved by accounting for the actual crystallographic texture of the specimen, and assuming pancake-shaped two-point statistics for the morphological texture. A discussion of the limitations of this approach, originally developed for bulk materials, is given for the specific case of thin films.

A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.

Mechanical field fluctuations in polycrystals estimated by homogenization techniques

The fluctuation of mechanical fields arising in polycrystals is investigated. These materials are viewed as composites of the Hashin–Shtrikman type with a large number of anisotropic phases and a ‘granular’ topology. We show that the estimation of the intra–phase stress and strain (rate) second moments comes down to the resolution of a linear system of equations. Applied to a linear viscous face–centred cubic (FCC) polycrystal, it is observed that significant local slip rates are estimated even when the corresponding Schmid factor vanishes, due to the intergranular interactions. For the application to viscoplastic polycrystals, the secant and affine nonlinear extensions of the self–consistent scheme are compared. At large stress sensitivity (n = 30), it is observed that the secant linearization leads to almost uniform slip rates for all slip systems in every phase, whereas the affine approach predicts a much larger spread. Furthermore, there is no one–to–one relation between the phase–average stress (or strain rate) and the corresponding second moment. It is emphasized that intra–phase strain–rate heterogeneities should be accounted for when dealing with microstructure evolution.

Thermal creep of Zr-Nb1%-O alloys: experimental analysis and micromechanical modelling

Abstract Zirconium alloys present a large variability of their mechanical behaviour with respect not only to their chemical composition but also to their microstructure. We analyze here the creep behaviour at 400 °C of two Zr–Nb1%–O alloys presenting identical chemical composition, crystallographic texture, grain size and grain shape. Both alloys only differ by the thermal cycles imposed during the fabrication process, either below (alloy A) or alternatively above and below (alloy B) the monotectoid transition. This sole difference gives rise to creep rates varying by a factor of about 4 between the two alloys. From a microstructural point of view, alloys A and B differ by the precipitates distribution and the thermodynamical state (alloy B is in a metastable equilibrium state). Our experimental analysis based on mechanical tests, transmission electron microscopy (TEM) observations and phase analysis by X-ray diffraction strongly suggests an hardening effect of Nb in solid solution to explain the differences between alloys A and B. This result is confirmed by TEM X-ray spectrometry which gives a weight content of Nb in solid solution differing by about 0.1% between the two alloys. A predictive micromechanical model, based on the self-consistent affine scheme, is then applied. This model well captures the anisotropy of the specimens, and describes accurately both transient and secondary creep regimes. As a result of the identification procedure, identical hardening laws are obtained for the two alloys at the grain scale, and the saturating reference stress for prismatic slip is found to be higher for alloy B by about 30 MPa with respect to alloy A.

Elastoviscoplastic micromechanical modeling of the transient creep of ice

A salient feature of the rheology of isotropic polycrystalline ices is the decrease of the strain rate by more than 2 orders of magnitude during transient creep tests to reach a secondary creep regime at a strain which is systematically of ∼1%. We use a recent (so-called “affine”) version of the self-consistent mean-field theory to model the elastoviscoplastic behavior of ice. The model aims at bridging scales between the rheology of single grain and the one of polycrystals by evaluating the intergranular interactions. It takes into account the long-term memory effects, which manifests itself by the fact that local stress and strain rate in grains depend on the whole mechanical history of the polycrystal. It is shown that the strong hardening amplitude during the transient creep is entirely explained by the stress redistribution within the specimen, from an almost uniform stress distribution upon instantaneous loading (purely elastic response) to strong interphase and intraphase heterogeneities in the stationary regime (purely viscoplastic response). The experimental hardening kinetic is much too slow to be explained by the same process; it is attributed to the hardening of hard glide slip systems (prismatic slip) in the transient regime. Moreover, the model very well reproduces the permanent creep rate of several highly anisotropic specimens of the Greenland Ice Core Project ice core (pronounced crystallographic textures), when accounting for a single-grain rheology that well matches the experimental one. Our results are consistent with recent findings concerning dislocation dynamics in ice.

Residual strain distribution in Zircaloy-4 measured by neutron diffraction and estimated by homogenization techniques

Abstract Neutron scattering provides a volume measurement of the elastic strain distribution in polycrystals. Phase average elastic strains have been measured on a plastically deformed specimen of Zircaloy-4, and an excellent agreement was obtained with the prediction of the affine self-consistent scheme for nonlinear elasto-viscoplasticity.

Asymptotic homogenization of periodic thermo-magneto-electro-elastic heterogeneous media

The asymptotic homogenization method is applied to a family of boundary value problems for linear thermo-magneto-electro-elastic (TMEE) heterogeneous media with periodic and rapidly oscillating coefficients. Using a matrix notation, the procedure for constructing the formal asymptotic solution is described. Two ways to validate the asymptotic analysis are explained. The main differences/similarities with respect to the asymptotic homogenization models reported in recent papers are remarked. The analytical expressions for effective coefficients of laminated media with any finite number of anisotropic TMEE layers are explicitly obtained via the matrix notation. Such formulae can be applied to investigate the global behavior of functionally graded TMEE multilayers. The important case of bilaminates composites with anisotropic homogeneous phases is also expressed in a compact form using matrices and vectors depending on the individual geometrical and mechanical properties of the components. The case of a bilaminate with homogeneous transversely isotropic TMEE layers is studied. A chain of equalities relating all thermal (thermoelastic, pyroelectric, pyromagnetic and heat capacity) effective coefficients was found for the example corresponding to a parallel connectivity. An analytical formula to estimate the volume fraction for which the pyroelectric and pyromagnetic effects realize their extreme values is given. Comparisons with recently published results are included.

Elastic behavior of polycrystalline thin films inferred from in situ micromechanical testing and modeling

Synchrotron x-ray diffraction combined with in situ tensile testing is used to investigate the anisotropic elastic behavior of gold thin films exhibiting a fiber texture and columnar grains. Micromechanical modeling based on the self-consistent model and accounting for both crystallographic and morphological textures is applied. Flat-disk shaped inclusions must be used in the model to reproduce accurately experimental data, owing to the surface effects.

Full-Field vs. Homogenization Methods to Predict Microstructure-Property Relations for Polycrystalline Materials

In this chapter, we review two recently proposed methodologies, based on crystal plasticity, for the prediction of microstructure–property relations in polycrystalline aggregates. The first, known as the second-order viscoplastic self-consistent (SC) method, is a mean-field theory, while the second, known as the fast Fourier transform (FFT)-based formulation, is a full-field method. The main equations and assumptions underlying both formulations are presented, using a unified notation and pointing out their similarities and differences. Concerning mean-field SC homogenization theories for the prediction of mechanical behavior of nonlinear viscoplastic polycrystals, we carry out detailed comparisons of the different linearization assumptions that can be found in the literature. Then, after validating the FFT-based full-field formulation by comparison with available analytical results, the effective behavior of model material systems predicted by means of different SC approaches are compared with ensemble averages of full-field solutions. These comparisons show that the predictions obtained by means of the second-order SC approach– which incorporates statistical information at grain level beyond first-order, through the second moments of the local field fluctuations inside the constituent grains– are in better agreement with the FFT-based full-field solutions. This is especially true in the cases of highly heterogeneous materials due to strong nonlinearity or single-crystal anisotropy. The second-order SC approach is next applied to the prediction of texture evolution of polycrystalline ice deformed in compression, a case that illustrates the flexibility of this formulation to handle problems involving materials with highly anisotropic local properties. Finally, a full three-dimensional implementation, the FFT-based formulation, is applied to study subgrain texture evolution in copper deformed in tension, with direct input and validation from orientation images. Measurements and simulations agree in that grains with initial orientation near tend to develop higher misorientations. This behavior can be explained in terms of attraction toward the two stable orientations and grain interaction. Only models like the FFT-based formulation that account explicitly for interaction between individual grains are able to capture these effects.

A self-consistent estimate for linear viscoelastic polycrystals with internal variables inferred from the collocation method

The correspondence principle is customarily used with the Laplace–Carson transform technique to tackle the homogenization of linear viscoelastic heterogeneous media. The main drawback of this method lies in the fact that the whole stress and strain histories have to be considered to compute the mechanical response of the material during a given macroscopic loading. Following a remark of Mandel (1966 Mecanique des Milieux Continus(Paris, France: Gauthier-Villars)), Ricaud and Masson (2009 Int. J. Solids Struct. 46 1599–1606) have shown the equivalence between the collocation method used to invert Laplace–Carson transforms and an internal variables formulation. In this paper, this new method is developed for the case of polycrystalline materials with general anisotropic properties for local and macroscopic behavior. Applications are provided for the case of constitutive relations accounting for glide of dislocations on particular slip systems. It is shown that the method yields accurate results that perfectly match the standard collocation method and reference full-field results obtained with a FFT numerical scheme. The formulation is then extended to the case of time- and strain-dependent viscous properties, leading to the incremental collocation method (ICM) that can be solved efficiently by a step-by-step procedure. Specifically, the introduction of isotropic and kinematic hardening at the slip system scale is considered.