Composites with auxetic inclusions showing both an auxetic behavior and enhancement of their mechanical properties
Introduction
In recent years, many attempts to conceive or produce new auxetic materials and structures have led to the identification of structures endowed with negative Poisson’s ratio, including metals having a cubic unit cell when loaded in the (1 1 0) direction [1], silicates [2], [3], [4], [5], [6], [7], [8], [9], or zeolites [6], [7], [8], [9]. However, relatively few composites with negative Poisson’s ratio have been manufactured and characterized; only recent works present auxetic composites made of intrinsically auxetic phases [10], in this last work composites of low modulus made from an auxetic yarn in a woven textile.
Auxeticity (a denomination adopted to characterize materials and structured having an overall negative Poisson’s ratio) has now been identified in a wide range of man-made materials, including foams, liquid crystalline polymers and micro/nano-structured polymers [11], [12], [13], [14], [15], [16]. Auxetic materials present some unique properties in comparison to common materials [17], since they show enhanced mechanical properties such as hardness, indentation, shear and fracture resistance. Some of the viscoelastic properties and the deformation behavior are also shown to be enhanced by auxeticity [18]. Thanks to static and free-vibration simulations of sandwich beams with different core cellular materials, Scarpa et al. [19] obtained both an enhanced stiffness per unit weight and increased modal loss factors, using two-phase cellular solids with a re-entrant skeleton. It is accordingly thought that composites incorporating auxetic inclusions have a great potential in terms of enhanced properties.
This work is centered on the concept of auxetic composites made of an embedded auxetic inclusion into an elastic matrix. The main novelty advocated in this contribution is the consideration of architectured inclusions having a lattice topology, leading to a negative Poisson’s ratio of the composite. The auxetic behavior is related to a very specific microstructure or nanostructure endowed with specific deformation mechanisms: for instance, the re-entrant honeycomb [20], [21] has been identified as one typical microstructure exhibiting the NPR (this shortcut for negative Poisson’s ratio will be used here and in the sequel, with PR meaning Poisson’s ratio). The chiral honeycomb structure [22], the hexachiral, tetra-antichiral and rotachiral lattices [23], are further typical well-known architectured model materials exhibiting the NPR effect.
In the present paper, the theoretical construction of new classes of auxetic composite materials is considered, based on ellipsoidal and circular inclusions endowed with a negative Poisson’s ratio [24]. The inclusions have a discrete topology consisting of a network of beams, and the matrix behavior is here restricted to be elastic and isotropic. A few researchers have focused on this idea and methodology by considering virtual or idealized NPR’s inclusion [25], [26], [27], without taking care of whether or not such a virtual inclusion could be obtained in reality and how, especially in terms of a corresponding microstructure.
The main objective of this work is to provide a quantitative understanding of the impact of the architecture (lattice topology) and micromechanical properties of the inclusion on the overall mechanical properties of the composite, and to develop appropriate and accurate micromechanical models that can be used in a predictive manner.
As a first step, the discrete asymptotic homogenized technique (the short cut DAH will be employed in the sequel) will be involved to derive the expressions (in closed form in the considered linear framework) of the equivalent moduli of architectured inclusion materials (the last coinage will be used to denote a micro-structure consisting of a quasi periodical array of microscopic beams), vs. the geometrical and mechanical micro-parameters of the underlying lattice. In a second step, a numerical homogenization technique based on the FE method will be employed for the computation of the overall properties of the composite material.
This contribution is organized as follows: a survey of the DAH method and the determination of the effective properties of architectured inclusions shall be exposed in Section 2. The overall properties of the resulting composite material and numerical results will be obtained in Section 3 from numerical homogenization. A summary of the main results is presented in the conclusion (Section 4).
Regarding notations, tensors are denoted using boldface symbols.
Section snippets
Effective properties of architectured inclusions
The overall properties of architectured lattices or systems have been investigated thanks to several homogenization techniques, amongst which discrete homogenization proves useful when considering discrete structures from the onset, such as beam lattices. The DAH method is a mathematical technique to derive the equivalent continuous medium behavior of a quasi periodical discrete structure made of the repetition of an elementary basic cell. This technique is inspired from the homogenization of
Numerical homogenization and applications
The idea of using the concept of inclusion embedded in a surrounding matrix for predicting the effective properties of an inhomogeneous continuum such as a composite material has been initially introduced in the pioneering contribution of Eshelby [1]. In the literature, the determination of the overall elastic properties of composites with ellipsoidal shaped inclusions is based on the averaged stress, strain and elastic-energy fields [38]. Those global fields are related to the local fields in
Conclusion and perspectives
In this contribution, a novel concept of composites made of inherently auxetic inclusions consisting of beam lattices has been proposed, showing overall both improved mechanical properties (moduli) and negative Poisson’s ratio in certain configurations. A two-steps micromechanical approach for the calculation of the effective mechanical properties of the composite has been adopted: the discrete asymptotic homogenization allows the calculation of the effective mechanical properties of the
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