Julien Yvonnet

The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains

This paper presents a new multi-scale method for the homogenization analysis of hyperelastic solids undergoing finite strains. The key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method, in order to alleviate the complexity of multiscale procedures, which usually involve a large number of nonlinear nested problems to be solved. The problem associated with the representative volume element (RVE) is solved via a model reduction method (proper orthogonal decomposition). The reduced basis is obtained through pre-computations on the RVE. The technique, coined as reduced model multiscale method (R3M), allows reducing significantly the computation times, as no large matrix needs to be inverted, and as the convergence of both macro and micro problems is enhanced. Furthermore, the R3M drastically reduces the size of the data base describing the history of the micro problems. In order to validate the technique in the context of porous elastomers at finite strains, a comparison between a full and a reduced multiscale analysis is performed through numerical examples, involving different micro and macro structures, as well as different nonlinear models (Neo-Hookean, Mooney-Rivlin). It is shown that the R3M gives good agreement with the full simulations, at lower computational and data storage requirements.

A critical comparison of several numerical methods for computing effective properties of highly heterogeneous materials

Modelling transport and long-term creep in concrete materials is a difficult problem when the complexity of the microstructure is taken into account, because it is hard to predict instantaneous elastic responses. In this work, several numerical methods are compared to assess their properties and suitability to model concrete-like microstructures with large phase properties contrast. The methods are classical finite elements, a novel extended finite element method (@m-xfem), an unconstrained heuristic meshing technique (amie), and a locally homogenising preprocessor in combination with various solvers (benhur). The benchmark itself consists of a number of simple and complex microstructures, which are tested with a range of phase contrasts designed to cover the needs of creep and transport modelling in concrete. The calculations are performed assuming linear elasticity and thermal conduction. The methods are compared in term of precision, ease of implementation and appropriateness to the problem type. We find that xfem is the most suitable when the mesh if coarse, and methods based on Cartesian grids are best when a very fine mesh can be used. Finite element methods are good compromises with high flexibility.

First-principles based multiscale model of piezoelectric nanowires with surface effects

A continuum model of nanowires incorporating surface piezoelectricity is proposed which extends the electric enthalpy energy with surface terms. The corresponding equations are solved by a numerical method using finite elements technique. A methodology is introduced to compute the surface piezoelectric coefficients by first-principles calculations through the Berry phase theory. We provide the e33s, e31s, and e15s piezoelectric coefficients of (101¯0) surfaces for hexagonal wurtzite nanowires made of GaN, ZnO, and AlN. The effective piezoelectric coefficient along the axis of the nanowire is found to increase when the diameter decreases, for the three studied materials. Finally, the solution of the continuum model is compared with large-size first-principles calculations on piezoelectric nanowires.

On the choice of parameters in the phase field method for simulating crack initiation with experimental validation

The phase field method is a versatile simulation framework for studying initiation and propagation of complex crack networks without dependence to the finite element mesh. In this paper, we discuss the influence of parameters in the method and provide experimental validations of crack initiation and propagation in plaster specimens. More specifically, we show by theoretical and experimental analyses that the regularization length should be interpreted as a material parameter, and identified experimentally as it. Qualitative and quantitative comparisons between numerical predictions and experimental data are provided. We show that the phase field method can predict accurately crack initiation and propagation in plaster specimens in compression with respect to experiments, when the material parameters, including the characteristic length are identified by other simple experimental tests.

Towards an elastic model of wurtzite AlN nanowires

Starting with ab initio calculations of AlN wurtzite [0001] nanowires with diameters up to 4 nm, a finite element method is developed to deal with larger nanostructures/nanoparticles. The ab initio calculations show that the structure of the nanowires can be well represented by an internal part with AlN bulk elastic properties, and one atomic surface layer with its own elastic behavior. The proposed finite element method includes surface elements with their own elastic properties using surface elastic coefficients deduced from the ab initio calculations. The elastic properties obtained with the finite element model compare very well with those obtained with the full ab initio calculations.

Characterization of surface and nonlinear elasticity in wurtzite ZnO nanowires

Surface elasticity and nonlinear effects are reported in ZnO nanowires and characterized by ab initio calculations. Fully anisotropic elastic and stress coefficients related to (101¯0) surfaces are provided and used to construct a continuum model of nanowires based on the Gurtin-Murdoch surface elasticity theory, able to capture mechanical size effects. Nonlinear elasticity is observed through non-zero third order energy derivative terms with respect to axial strain in the direction of the nanowire. The associated material parameters are found to be themselves size-dependent.

Multiscale modeling of nonlinear electric conductivity in graphene-reinforced nanocomposites taking into account tunnelling effect

Tunnelling effect is a possible mechanism to explain the apparent large electric conductivity and nonlinear electric behavior of graphene-reinforced nanocomposites with polymer matrix. In this work, a numerical modeling framework is proposed to evaluate the effective electric conductivity in polymer composites reinforced with graphene sheets, taking into account the electrical tunnelling effect, which allows conduction between graphene sheets at nanometric distances. We introduce a nonlinear Finite Element formulation and a numerical methodology to model the nonlocal and nonlinear effects introduced by the tunnelling effect conduction model within the polymer matrix between close graphene sheets. In addition, to avoid meshing the thickness of the graphene sheets and in view of their very high aspect ratio, a highly conducting surface model is employed. The computed effective conductivity is evaluated over representative volume elements containing arbitrary distributed graphene sheets. The results exhibit tendencies and percolation thresholds which are in qualitative agreement with the available experimental results.

A non-concurrent multiscale method for computing the response of hyperelastic heterogeneous structures

We propose a new numerical method for computing the response of structures made of heterogeneous nonlinear elastic materials. The first step is to define a representative volume element (r.v.e.) associated to the microstructure. Then, the effective potential, or the overall strain density function, is computed numerically for a finite set of points in the macroscopic strains space. In the computation of structure, stress and tangent stiffness tensors can be obtained through interpolation and derivation in the discrete set of potential values. Material properties contrast, anisotropy and morphology of microstructure are arbitrary.

Bubble and Hermite Natural Element Approximations

In this paper, new natural element approximations are proposed, in order to address issues associated with incompressibility as well as to increase the accuracy in the Natural Element Method (NEM). The NEM exhibits attractive features such as interpolant shape functions or auto-adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well known inf-sup, or LBB condition. The first proposed approach constructs richer NEM approximation schemes by means of bubbles associated with the topological entities of the underlying Delaunay tessellation, allowing to pass the LBB and to remove pressure oscillations in the incompressible limit. Despite of its simplicity, this approach does not construct approximation with higher order consistency. The second part of the paper deals with a discussion on the construction of second-order accurate NEM approximations. For this purpose, two techniques are investigated : (a) the enrichment in the MLS framework of the bubbles with higher-order polynomials and (b) the use of a new Hermite-NEM formulation.

Low electrical percolation thresholds and nonlinear effects in graphene-reinforced nanocomposites: A numerical analysis

A numerical model of graphene-reinforced nanocomposites taking into account the electric tunneling effect is employed to analyze the influence of microstructural parameters on the effective electric conductivity and the percolation thresholds of the composite. The generation procedure for the random microstructures of graphene-reinforced nanocomposites is described. Effects of the barrier height, of graphene aspect ratio and alignment of graphene sheets have been quantitatively evaluated. The results show that both higher graphene aspect ratio and lower barrier height can lead to smaller percolation threshold, and the alignment of graphene sheets results in anisotropic electrical behavior without affecting the percolation threshold. The numerical model also shows the importance of the tunneling effect to reproduce the nonlinear electric behavior and the low percolation thresholds reported in the literature. Finally, results are compared with available experimental data.