Evolution du bilan énergétique dans les matériaux métalliques sous sollicitation cyclique.

In the case of cyclic plasticity, the validity of a constitutive model is usually assessed using stress-strain curves. However, this description can be enriched by adopting an energetic point of view. Thus, in the present work, a multiscale model is developed to estimate the amount of energy which is either dissipated into heat or stored in the material in a medium carbon steel under cyclic loading. The results emphasize the heterogeneous aspect of the stored and dissipated energy fields at a microscopic scale

Experimental and numerical study of the evolution of stored energy in metallic materials under cyclic loading

In-service loading conditions usually generate complex cyclic stress states. As such, the choice of an appropriate multiaxial fatigue criterion plays a crucial role in obtaining correct fatigue predictions. In the case of high cycle fatigue, the observation of the stabilized behavior is generally required to build either stress-based criteria or energy-based criteria. ...

A non-local model for the description of twinning in polycrystalline materials at moderate strains: application to a magnesium alloy

A polycrystalline plasticity model, which incorporates the contribution of deformation twinning, is proposed. For this purpose, each material point is treated as a composite material consisting of a parent constituent and multiple twin variants. In the constitutive equations, the twin volume fractions and their spatial gradients are treated as external state variables to account for the contribution of twin boundaries to free energy. The set of constitutive relations is implemented in a spectral solver, which allows solving the differential equations resulting from equilibrium and compatibility conditions. The proposed model is then used to investigate the behavior of a AZ31 magnesium alloy. For the investigated loading conditions, the mechanical behavior is controlled by the joint contribution of basal slip and tensile twinning. Also, according to the numerical results, the development of crystallographic texture, morphological texture and internal stresses is consistent with the experimental observations of the literature

A comparison between different methods for the numerical simulation of polycrystalline aggregates

The macroscopic behavior of polycrystalline materials is largely influenced by the shape, the arrangement and the orientation of crystallites. Different methods have thus been developed to determine the effective behavior of such materials as a function of their microstructural features. In this work, which focuses on polycrystalline materials with an elastic-viscoplastic behavior, the self-consistent (SC) method [1], the finite element (FE) method and the spectral (FFT) method [2] are compared. These common methods are used to determine the effective behavior of different 316L polycrystalline aggregates subjected to various loading conditions (uniaxial tension, cyclic tension/compression).(...

A thermodynamically consistent formulation of the Johnson–Cook model

The model of Johnson and Cook, which includes a viscoplastic flow rule and a damage criterion, is widely used to describe the mechanical behaviour of metallic materials subjected to severe loading conditions, such as those encountered during fabrication operations or impact. This model has been built on empirical, rather than physical, grounds. The present paper therefore aims at revisiting the model of Johnson and Cook from the view point of thermodynamics with internal variables. The interest of this approach is twofold. First, it provides a guide for the construction of a complete thermomechanical constitutive model, with some constitutive relations not only for the stress tensor but also specific internal energy, specific entropy and heat flux vector. Second, it allows highlighting some possible limitations of the original model of Johnson and Cook. Such limitations can be circumvented with an alternative model, which is described in the present work. For illustration purpose, some applications of both the original and alternative models are presented in the final section

Constitutive equations for thermo-elasto-plastic metallic materials undergoing large temperature variations

In the present work, a framework for the development of constitutive models for metallic materials in a thermomechanical context is proposed. Such a framework provides some guidelines to deal with processes involving large temperature variations, which is typical of manufacturing operations or severe service situations. The proposed framework relies on the additive decomposition of the logarithmic strain tensor to include the contributions of elasticity, plasticity and thermal expansion to deformation. Also, the classical internal variable concept is used to describe the history effects (e.g., hardening and recovery) associated with the development of plasticity. Particular attention is given to considering the impact of temperature on thermophysical properties. For the purpose of illustration, the proposed framework is used to build a constitutive model for polycrystalline copper. The resulting set of constitutive equations allows investigating the thermomechanical behavior of this specific material for some simple deformation histories. The corresponding results are finally used to evaluate the temperature-dependence of common thermophysical properties. The importance of the different heat sources that contribute to the heat diffusion equation is also discussed

A non-local damage model for the fatigue behaviour of metallic polycrystals

The development of fatigue damage in metallic materials is a complex process influenced by both intrinsic (e.g. texture, defects) and extrinsic (e.g. loading mode, frequency) factors. To better understand this process, some efforts have been made to develop microstructure-sensitive models that consider the impact of microstructural heterogeneities on the formation of fatigue cracks. An important limitation of such models is their inability to describe the transition between the nucleation and growth stages in a consistent thermodynamic framework. To circumvent this limitation, a constitutive model, which is appropriate for the description of both nucleation and early growth, is proposed. For this purpose, non-local damage mechanics is used to construct a set of constitutive relations that explicitly considers the progressive stiffness reduction due to the formation of fatigue cracks. Specifically, a damage variable is attached to each slip system, which allows considering both the anisotropic aspect of fatigue damage and closure effects. The spatial gradients of the damage variables are treated as additional state variables to account for the increase of surface energy associated with the formation of fatigue cracks. For the evolution equations to be compatible with the second law of thermodynamics, a modified form of the entropy production inequality is adopted. For illustration purposes, the non-local damage model is implemented in a spectral solver. Some numerical examples are then presented. These examples allow discussing the ability of the proposed model to describe the impact of loading conditions and pre-existing defects on the fatigue behaviour of metallic polycrystals

The dependence of X-ray elastic constants with respect to the penetration depth

X-ray diffraction techniques are widely used to estimate stresses within polycrystalline materials. The application of these techniques requires the knowledge of the X-ray elastic constants relating the lattice strains to the stress state. Different analytical methods have been proposed to evaluate the X-ray elastic constants from the single-crystal elastic constants. For a given material, such methods provide the bulk X-ray elastic constants but they do not consider the role of free surfaces. However, for many practical applications of X-ray diffraction techniques, the penetration depth of X-rays is the same order of magnitude as the grain size, which means that the influence of the free surface on X-ray elastic constants cannot be excluded. In the present work, a numerical procedure is proposed to evaluate the surface and bulk X-ray elastic constants of polycrystalline materials. While the former correspond to the situation where the penetration is infinitely small in comparison with the grain size, the latter are representative of an infinite penetration depth with no free-surface effect. According to numerical results, the difference between surface and bulk X-ray elastic constants is important for strongly anisotropic crystals. Also, it is possible to propose a relation that allows evaluating X-ray elastic constants as a function of the ratio between the penetration depth and the average grain size. The corresponding parameters of such a relation are provided here for many engineering materials

Thermodynamic framework for variance-based non-local constitutive models

The present work deals with the development of a framework dedicated to the construction of constitutive models with non-local internal variables. Such internal variables allow considering the impact of microstructural changes on the current state of a material point. Non-locality is introduced by considering, not only the spatial average, but also the spatial variance of an internal variable in constitutive relations. The proposed framework relies on continuum thermodynamics to construct the set of constitutive equations. Such a framework allows including some information regarding the spatial distribution of internal variables when constructing non-local models for thermomechanical applications. In contrast with gradient-type models, this strategy does not require additional equilibrium equations and boundary conditions. For the purpose of illustration, some numerical examples are presented. According to the numerical results, the proposed framework can be used to circumvent the difficulties associated with excessive spatial localization or to consider size effects

Self-consistent modelling of heterogeneous materials with an elastic-viscoplastic behavior: Application to polycrystalline agregates

The self-consistent scheme is a common homogenization method that was developed to connect local deformation mechanisms to the overall behavior of heterogeneous disordered materials. In the past decades, many efforts have been made to obtain extensions of the self-consistent approximation to the non-linear case. This work focuses on the specific case of heterogeneous materials with an elastic-viscoplastic behavior. For such materials, the overall behavior is strongly dependent on the space-time couplings originating from the differential form of the local constitutive law. Different approaches have thus been developed to describe the impact of such complex couplings on the overall behavior. (...