Gradient Theory: Constitutive Models and Fatigue Criteria; Application in Micromechanics

D.H. Luu 1
1 Micro-Behavior, Micro-Fatigue
LMS - Laboratoire de mécanique des solides
Abstract : In the present thesis, two new classes of phenomenological models in the framework of the continuum thermodynamics and gradient theory are proposed. The first one is standard gradient constitutive model used to deal with the mechanical problems at micro-scale, and the other concerns gradient fatigue criteria for the problems at small scale. Using these, some common effects which are not captured yet in the classical mechanics but become significant at sufficiently small scales, are taken into account. For each class, the size and gradient effects which are the two effects most commonly discussed and very confused between each other in the literature, are clearly distinct and demonstrated to be integrated into the later via gradient terms. The thesis contains two principal contents presented in the part A and part B, respectively corresponding to the two new model classes. The following are their summary: Part A- Standard Gradient Constitutive Models: Application in Micro-Mechanics. A formulation of Standard Gradient Plasticity Models, based on an abundant researches on strain gradient plasticity (SPG) theory in the literature such as the ones of Q.S. Nguyen (2000, 2005, 2011 and 2012), is proposed and numerically implemented. The models are based on a global approach in the framework of continuum thermodynamics and generalized standard materials where the standard gradients of the internal parameters in the set of state variables are introduced. The governing equations for a solid are derived from an extended version of the virtual work equation (Frémond, 1985 or Gurtin, 1996). These equations can also be derived from the formalism of energy and dissipation potentials and appear as a generalized Biot equation for the solid. The gradient formulation established in such way is considered a higher-order extension of the local plasticity theory, with the introduction of the material characteristic length scale and the insulation boundary condition proposed by Polizzotto. The presence of strain gradient leads to a Laplacian equation and to non-standard boundary value problem with partial differential equations of higher order. A computational method, at the global level, based on diffusion like-problem spirit is used. Illustrations are given and applied to some typical problems in micro-mechanics to reproduce the well-known mechanical phenomenon, the effect "smaller is stronger". A good agreement between numerical results and reference counterparts is found; mesh-independence of numerical results is observed. Part B- Gradient Fatigue Criteria at Small Scale. A reformulation of gradient fatigue criteria is proposed in the context of multiaxial high-cycle fatigue (HCF) of metallic materials, initiated by Papadopoulos 1996. The notable dependence of fatigue limit on some common factors concerning the material specimen size is analysed and modeled. These factors, which are not taken into account before in classical fatigue criteria but become significant at sufficiently small scales, are included in the new formulation. Among such factors, three ones intimately related to each other, the pure size (smaller is stronger), stress gradient (higher gradient is higher resistance) and loading (i.e. loading mode) effects, are here investigated. An effort has been made to roughly integrate all these effects into only one through gradient terms. According to that, a new class of fatigue criteria with stress gradient terms introduced not only in the normal stress but also in the shear stress parts, are formulated. Such a formulation allows to capture all the pure size (if important) and stress gradient (if any) effects, as well as to cover a wide range of loading effect (traction, bending and shearing, for instance). Due to such a property, these new criteria are naturally generalized to multiaxial loadings to be a new version of stress gradient dependent multiaxial fatigue criteria. Application to some classical fatigue criteria such Crossland and Dang Van is provided as illustrations. As shown, classical fatigue criteria as well as the one of Papadopoulos 1996, are considered special cases of the new respective criteria. An overview for the whole thesis is put in this Summary, and an overview for each model class is found in the Chapter 1 where a general introduction of the thesis is given. Their corresponding detail are presented in the Chapters 2-4 (for part A) and Chapters 5-6 (for part B). The last chapter, Chapter 7, is dedicated to general conclusions and perspectives.
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D.H. Luu. Gradient Theory: Constitutive Models and Fatigue Criteria; Application in Micromechanics. Mechanics of the solides [physics.class-ph]. Ecole Polytechnique X, 2013. English. ⟨pastel-00866081⟩

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