Coupled phase field and nonlocal integral elasticity analysis of stress-induced martensitic transformations at the nanoscale: boundary effects, limitations and contradictions

In this paper, the coupled phase field and local/nonlocal integral elasticity theories are used for stress-induced martensitic phase transformations (MPTs) at the nanoscale to investigate the limitations and contradictions of the nonlocal integral elasticity, which are due to the fact that the support of the nonlocal kernel exceeds the integration domain, i.e., the boundary effect. Different functions for the nonlocal kernel are compared. In order to compensate the boundary effect, a new nonlocal kernel, i.e., the compensated two-phase kernel, is introduced, in which a local part is added to the nonlocal part of the two-phase kernel to account for the boundary effect. In contrast to the previously introduced modified kernel, the compensated two-phase kernel does not lead to a purely nonlocal behavior in the core region, and hence no singular behavior, and consequently, no computational convergence issue is observed. The nonlinear finite element approach and the COMSOL code are used to solve the coupled system of Ginzburg–Landau and local/nonlocal integral elasticity equations. The numerical implementation of the phase field-local elasticity equations and the 2D nonlocal integral elasticity are verified. Boundary effect is investigated for MPT with both homogeneous and nonhomogeneous stress distributions. For the former, in contrast to the local elasticity, a nonhomogeneous phase transformation (PT) occurs in the nonlocal case with the two-phase kernel. Using the compensated two-phase kernel results in a homogeneous PT similar to the local elasticity. For the latter, the sample transforms to martensite except the adjacent region to the boundary for the local elasticity, while for the two-phase kernel, the entire sample transforms to martensite. The solution of the compensated two-phase kernel, however, is very similar to that of the local elasticity. The applicability of boundary symmetry in phase field problems is also investigated, which shows that it leads to incorrect results within the nonlocal integral elasticity. This is because when the symmetric portions of a sample are removed, the corresponding nonlocal effects on the remaining portion are neglected and the symmetric boundaries violate the normalization condition. An example is presented in which the results of a complete model with the two-phase kernel are different from those of its one-fourth model. In contrast, the compensated two-phase kernel can generate similar solutions for both the complete and one-fourth models. However, in general, none of the nonlocal kernels can overcome this issue. Therefore, the symmetrical models are not recommended for nonlocal integral elasticity based phase field simulations of MPTs. The current study helps for a better study of nonlocal elasticity based phase field problems for various phenomena such as various PTs

Modélisation du comportement dissipatif d'un renfort de composite à fibres continues

L'objectif principal est de déterminer un modèle décrivant le comportement dissipatif d'un renfort de composite à fibres continues soumis à de grandes déformations. Compte ? tenu des caractéristiques d'inextensibilités des fibres constituant le composite, le mode de dissipation suit donc une cinématique de cisaillement simple. Un essai de Picture Frame étant réalisé, il est alors observable que lors de la décharge, la contrainte décrit un retour par un trajet asymptotique. Plusieurs modèles existent pour approcher ce comportement asymptotique mais ne peuvent pas être appliqués dans ce cas particulier sans réduire la simplicité du modèle proposé. En effet, l'unique inconnue du modèle est l'angle de cisaillement plastique. Le couplage entre la décomposition de Green ? Naghdi et de Kröner ? Lee permet de définir l'ensemble des tenseurs fondamentaux uniquement dépendant de la configuration initiale qui est donc connue. Ainsi, pour conserver cette simplicité et décrire ce retour asymptotique, la théorie des dérivées fractionnaires, le modèle proposé par Prager ou encore Popov ne sont pas adaptés. L'objectif est donc d'utiliser le principe du modèle de Mroz en écrivant des lois d'évolutions uniquement dépendantes à l'angle de cisaillement plastique

Hypoelastic, hyperelastic, discrete and semi-discrete approaches for textile composite reinforcement forming

International audienceThe clear multi-scale structure of composite textile reinforcements leads to develop continuous and discrete approaches for their forming simulations. In this paper two continuous modelling respectively based on a hypoelastic and hyperelastic constitutive model are presented. A discrete approach is also considered in which each yarn is modelled by shell finite elements and where the contact with friction and possible sliding between the yarns are taken into account. Finally the semi-discrete approach is presented in which the shell finite element interpolation involves continuity of the displacement field but where the internal virtual work is obtained as the sum of tension, in-plane shear and bending ones of all the woven unit cells within the element. The advantages and drawbacks of the different approaches are discussed

Maximum mechano-damage power release-based phase-field modeling of mass diffusion in damaging deformable solids

We formulate in three space dimensions a phase-field theory of mass diffusion in a damaging deformable solid matrix without making use of thermal quantities. The approach relies on three fundamental postulates: the diffusing species mass balance, the maximum mechano-damage power release principle, and an energy imbalance inequality. The solid is modelled mechanically as a Cauchy continuum. The kinematics of the continuum is given in terms of the diffusing species concentration and the solid matrix placement and damage fields. The formulation is based on the multiplicative decomposition of the total deformation gradient into an intercalation and an elastic deformation gradients. Constitutively, the model relies on three free energy functionals, i.e., chemical, damage, and strain energies. Damage is the phase field of the formulation, as non-locality is ensured by the dependence of the damage energy upon the damage gradient. Aimed at giving an insight into the properties of the model, the results of finite-element dynamic simulations are reported for a mechanically confined one-dimensional continuum

Tension locking in finite-element analyses of textile composite reinforcement deformation

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Modeling of hyperelastic bending of fibrous media using second-gradient isogeometric analysis: Weaving and braiding applications

International audienceFor many applications, numerical modeling of composite reinforcements may require a detailed knowledge of the structure at the scale of the yarn, here called the mesoscopic scale. A numerical description of the yarn can be obtained using beam finite elements, shells, and most often three-dimensional (3D) solid elements. For a reliable description of the geometry, it is necessary to mesh with a large number of elements. With Non Uniform Rational B-Splines (NURBS) interpolation functions, on the other hand, not only is it possible to approximate the yarn geometry with a reduced number of degrees of freedom, but the high-order continuity allows a description of higher deformation gradient mechanics. These two points are exploited here, for a detailed presentation of the hyperelastic formalism considering fiber bending, and its implementation in an iso-geometric framework. Practical examples of weaving and braiding demonstrate the relevance of the formulation

Modelling the development of defects during composite reinforcements and prepreg forming

This article is part of the themed issue ‘Multiscale modelling of the structural integrity of composite materials’.International audienceDefects in composite materials are created during manufacture to a large extent. To avoid them as much as possible, it is important that process simulations model the onset and the development of these defects. It is then possible to determine the manufacturing conditions that lead to the absence or to the controlled presence of such defects. Three types of defects that may appear during textile composite reinforcement or prepreg forming are analysed and modelled in this paper. Wrinkling is one of the most common flaws that occur during textile composite reinforcement forming processes. The influence of the different rigidities of the textile reinforcement is studied. The concept of ‘locking angle’ is questioned. A second type of unusual behaviour of fibrous composite reinforcements that can be seen as a flaw during their forming process is the onset of peculiar ‘transition zones’ that are directly related to the bending stiffness of the fibres. The ‘transition zones’ are due to the bending stiffness of fibres. The standard continuum mechanics of Cauchy is not sufficient to model these defects. A second gradient approach is presented that allows one to account for such unusual behaviours and to master their onset and development during forming process simulations. Finally, the large slippages that may occur during a preform forming are discussed and simulated with meso finite-element models used for macroscopic forming

The difficulties in modeling the mechanical behavior of textile composite reinforcements with standard continuum mechanics of Cauchy. Some possible remedies

International audienceIn most cases, fibrous composite reinforcements can be modeled as a continuum medium. Nevertheless, it is shown on some examples that the models in the standard continuum mechanics of Cauchy are not able to correctly describe the mechanical behavior of fibrous reinforcements. The reasons for these difficulties are analyzed thanks to a very simplified model. The inability of the model of Cauchy to describe both the possibility of slippage between the fibres and the bending stiffness of the fibres is highlighted. Two approaches are proposed to overcome these difficulties and to model the mechanical behavior of fibrous reinforcements taking into account the local fibre bending stiffness. First, the continuum model is completed by a generalized second gradient continuum theory. An alternative solution is to add a stiffness related to the curvature to hexahedral finite elements. For this, the positions of neighboring elements are used to calculate the curvature in the element. In both cases, the local fibre stiffness is taken into account without modifying the weak shear stiffness

A nine nodes solid-shell finite element with enhanced pinching stress

International audienceIn this paper we present a low-order solid-shell element formulation—having only displacement degrees of freedom (DOFs), i.e., without rotational DOFs. The element has an additional middle node, that allows efficient and accurate analyses of shell structures using elements at extremely high aspect ratio. The formulation is based on the Hu–Washizu variational principle leading to a novel enhancing strain and stress tensor that renders the computation particularly efficient, with improved inplane and out-of-plane bending behavior (Poisson thickness locking). The middle-node is endowed with only one degree of freedom, in the thickness direction, allowing the assumption of a quadratic interpolation of the transverse displacement. Unlike solid-shell finite elements reported previously in the literature and formulated under the hypothesis of plane stress or with enhanced assumed strain parameter, the new solid-shell element here mentioned uses a complete three-dimensional constitutive law and gives an enhanced pinching stress, thanks to the middle-node. Moreover, to handle the various locking problems that usually arise on solid-shell formulation, the reduced integration technique is used as well as the assumed shear strain method. Finally to assess the effectiveness and performance of this new formulation, a set of popular benchmark problems, involving geometric non-linear analysis as well as elastic-plastic behavior has been investigated

Modélisation de ma mise en forme des renforts de composites tissés monoplis et multiplis

International audienceThe numerical simulation of composite forming permits to envisage the feasibility of a process without defect but also to know the directions of the reinforcements after shaping. These directions condition strongly the mechanical behaviour of the final textile composite structure. In addition, the angles between warp and weft yarns influence the permeability of the reinforcement and thus the filling of the resin in the case of a liquid moulding process. The forming of composite reinforcement can be made on a single ply or simultaneously on several plies. In this paper models for the single woven ply forming simulation are described. The semi-discrete approach is used for the simultaneous hemispherical forming of three layers.La simulation numérique de la mise en forme des composites permet d'analyser la faisabilité d'un procédé et ses défauts, mais aussi de déterminer les directions des renforts après la mise en forme. Ces directions conditionnement fortement le comportement mécanique de la pièce composite en service. La mise en forme peut concerner un seul pli ou bien plusieurs simultanément. Dans ce travail, les différentes approches actuellement utilisées pour la simulation de la mise en forme d'un pli de renfort tissé sont décrites. En particulier l'approche semi discrète est utilisée pour le formage hémisphérique simultané de trois plis