Qi-Chang He

Conewise linear elastic materials

Conewise linear elastic (CLE) materials are proposed as the proper generalization to two and three dimensions of one-dimensionalbimodular models. The basic elements of classical smooth elasticity are extended tononsmooth (or piecewise smooth) elasticity. Firstly, a necessary and sufficient condition for a stress-strain law to becontinous across the interface of the tension and compression subdomains is established. Secondly, a sufficient condition for the strain energy function to be strictlyconvex is derived. Thirdly, the representations of the energy function, stress-strain law and elasticity tensor are obtained fororthotropic, transverse isotropic andisotropic CLE materials. Finally, the previous results are specialized to apiecewise linear stress-strain law and it is found out that the pieces must be polyhedral convex cones, thus the CLE name.

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 more fundamental approach to damaged elastic stress-strain relations

Abstract For many engineering problems it can be assumed that the damaged material elastic response at a fixed damage state is linear and hyperelastic. With this assumption, a systematic and rigorous approach for formulating damaged elastic stress-strain relations is presented. The approach is based on the Fourier series representations of two naturally defined damage orientation distribution functions and on a theorem on elasticity tensors, and developed resorting neither to the notion of effective stress (strain) nor to the hypothesis of strain (stress) equivalence nor to that of elastic energy equivalence. The proposed approach is finally illustrated by applying it to initially isotropic materials and to unidirectional fiber-reinforced composites.

Matrix representations for 3D strain-gradient elasticity

The theory of first strain gradient elasticity (SGE) is widely used to model size and non-local effects observed in materials and structures. For a material whose microstructure is centrosymmetric, SGE is characterized by a sixth-order elastic tensor in addition to the classical fourth-order elastic tensor. Even though the matrix form of the sixth-order elastic tensor is well-known in the isotropic case, its complete matrix representations seem to remain unavailable in the anisotropic cases. In the present paper, the explicit matrix representations of the sixth-order elastic tensor are derived and given for all the 3D anisotropic cases in a compact and well-structured way. These matrix representations are necessary to the development and application of SGE for anisotropic materials.

The number and types of all possible rotational symmetries for flexoelectric tensors

Flexoelectricity is due to the electric polarization generated by a non-zero strain gradient in a dielectric material without or with centrosymmetric microstructure. It is characterized by a fourth-order tensor, referred to as flexoelectric tensor, which relates the electric polarization vector to the gradient of the second-order strain tensor. This paper solves the fundamental problem of determining the number and types of all possible rotational symmetries for flexoelectric tensors and specifies the number of independent material parameters contained in a flexoelectric tensor belonging to a given symmetry class. These results are useful and even indispensable for experimentally identifying or theoretically/numerically estimating the flexoelectric coefficients of a dielectric material.

Continuum mechanics modelling of large deformation contact with friction

Material forming, vehicle crash and human joints are but a few typical situations where large displacement contact mechanics plays a crucial role. Although classical computational methods combined with special constrained optimisation techniques have been successfully developed by many authors for solving such contact problems [see e.g. Kikuchi and Oden 1985], the underlying continuum mechanics formulation is lagging behind, probably due to its complexity. Most of the work is restricted to unilateral contacts without friction [Ciarlet and Necas, 1985; Ciarlet 1988; Curnier et al., 1992] and a tentative theory with friction is just emerging [Klarbring et al., 1991; Laursen and Simo, 1991, 1993; Klarbring, 1994].

Eshelby's tensor fields and effective conductivity of composites made of anisotropic phases with Kapitza's interface thermal resistance

Eshelbys results and formalism for an elastic circular or spherical inhomogeneity embedded in an elastic infinite matrix are extended to the thermal conduction phenomenon with a Kapitza interface thermal resistance between matrix and inclusions. Closed-form expressions are derived for the generalized Eshelby interior and exterior conduction tensor fields and localization tensor fields in the case where the matrix and inclusion phases have the most general anisotropy. Unlike the relevant results in elasticity, the generalized Eshelby conduction tensor fields and localization tensor fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of Kapitzas interface thermal resistance. With the help of these results, the size-dependent overall thermal conduction properties of composites are estimated by using the dilute, Mori–Tanaka, self-consistent and generalized self-consistent models. The analytical estimates are finally compared with numerical results delivered by the finite element method. The approach elaborated and results provided by the present work are directly applicable to other physically analogous transport phenomena, such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media, and to the mathematically identical phenomenon of anti-plane elasticity.

On the symmetries of 2D elastic and hyperelastic tensors

A thorough investigation is made of the independent point-group symmetries and canonical matrix forms that the 2D elastic and hyperelastic tensors can have. Particular attention is paid to the concepts relevant to the proper definition of the independence of a symmetry from another one. It is shown that the numbers of all independent symmetries for the 2D elastic and hyperelastic tensors are six and four, respectively. In passing, a symmetry result useful for the homogenization theory of 2D linear elastic heterogeneous media is derived.

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.

Solutions to Eshelby's problems of non-elliptical thermal inclusions and cylindrical elastic inclusions of non-elliptical cross section

Eshelby’s inclusion problem is solved for non-elliptical inclusions in the context of two-dimensional thermal conduction and for cylindrical inclusions of non-elliptical cross section within the framework of generalized plane elasticity. First, we consider a two-dimensional infinite isotropic or anisotropic homogeneous medium with a non-elliptical inclusion subjected to a prescribed uniform heat flux-free temperature gradient. Eshelby’s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite Laurent series. Next, we are interested in a three-dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non-elliptical cross section that undergoes uniform generalized plane eigenstrains. The solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti-plane strain part, exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, and combining the relevant results of Zou et al. (Zou et al. 2010 J. Mech. Phys. Solids 58, 346–372. (doi:10.1016/j.jmps.2009.11.008)) with those derived in the present work for Eshelby’s conduction tensor field and its area average.