A novel concept to develop composite structures with isotropic negative Poisson’s ratio: Effects of random inclusions

Abstract

Materials with negative Poisson’s ratio (NPR) effects have been studied for decades. However, the studies have mainly focused on 2D periodic structures which only have NPR effects in certain in-plane directions. In this paper, a novel concept is proposed to develop composite structures with isotropic NPR effects using NPR random inclusions. The study starts from a finite element analysis of deformation mechanisms of two 2D representative cells which are embedded with a re-entrant square and a re-entrant triangle, respectively. Based on the analysis results, the re-entrant triangles are selected as random inclusions into a matrix to form 2D composite structures. Four such composite structures are built with different numbers of inclusions through a parametric model, and their NPR effects and mechanical behaviors are analyzed using the finite element method. The results show that the isotropic NPR effects of composites can be obtained with high random re-entrant inclusions. Thus, the novel concept proposed is numerically proved by this study.

Introduction

Materials with a negative Poisson’s ratio (NPR) were first described in 1944 [1]. An intentional development of the materials with NPR was first published by Lakes in Science [2]. Since then, many efforts have been done to study the NPR effects and to fabricate materials with this nonconventional behavior. Since the NPR equips materials with increased shear modulus [3], [4], indentation resistance [3], [5], fracture toughness [6] and energy absorption ability [7], [8], there are multiple potential applications to improve engineering materials for textile, automotive, military, biomedical and aerospace engineering [3], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], etc.

At present, the man-made NPR materials are generally classified by their deformation mechanism and microstructure [15], e.g. re-entrant structure, polymeric structure, chiral structure, star-shape structure and other heterogeneous structures. However, the classification scheme conceals two important limitations of existing NPR materials: directions of NPR achieved and possibility of manufacturing and applications. To date, most of the NPR structures are studied in two-dimensional space, and the NPR effects are limited to certain directions. This is true, for example, for re-entrant, chiral, star-shape structures and rotating units. These structures have been well studied using skeletal models based on perfect rigid components with free pivots at mutual points of the components [17]. As most of the structures are periodic and often also symmetric [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], the NPR effect only occurs in certain in-plane directions, and the concept is hardly used in the case of three-dimensional structures. Up to now, there are only a few studies, which extend the NPR effects to 3D. Lakes [6] firstly reported a 3D foam structure with the NPR effect. Then he published a well-known idealized 3D re-entrant structure with a complicated tetrakaidecahedron structure [29]. Based on a 3D rotational structure, Alderson and Evans developed analytical models for tetrahedral framework silicates [30]. However, those studies have only focused on certain cellular materials with highly heterogeneous porous microstructures. This type of materials usually has low stiffness, which limits their applications. Besides, the developed theories are based on the complicated microstructures, and cannot be widely used for other engineering materials.

Another limitation of the previous and present researches on NPR structures is linked to their manufacture and applications. In addition to cellular solids and some fiber-reinforced composites, there are a few successful NPR materials which can be easily manufactured and employed in real engineering environment. A low-modulus composite consisting of NPR yarns was developed by Miller et al. [31], [32]. NPR angle-ply laminates could be easily produced with conventional materials and processing methods using special stacking sequence [33]. These composites have in-plane NPR effects under external loads. The advantage of these composites is their structural simplicity and deformation mechanism, in which the components and the overall composite can be produced using standard manufacturing techniques and commonly available materials. Although there are many potential applications, these structures are still limited due to their orientation-dependent NPR effects. Another applied NPR structure is a 3D graded structure developed as a vehicle blast-protector [7]. The 3D structure has inherent impact force mitigation effect due to its material-concentration effect under an impact loading. The graded structure is convenient for manufacturing due its structural simplicity. However, although it is a 3D structure, the NPR effect of the structure still occurs in one direction due to its periodicity.

As a result from the previous studies, the further development of NPR materials should more focus on disordered structures, which could equip the material with overall isotropic mechanical properties; and provide structures with a relatively simple deformation mechanism. Some efforts have been made to extend the effects of disordered structures to NPR materials. An optimized disordered grid structure has been analyzed by Horrigan et al. [34], which offers a new insight on the effect of randomness on the NPR effects of the overall material. Other examples with disordered structures are composites made with matrix and disordered reinforcing fibers/inclusions. Some composites have been proved to have NPR effects using both experimental and numerical methods [35], [36], [37], [38], [39], [40], [41], and some of them even have an out-of-plane NPR [36], [42], [43]. However, the present disordered structures can only have the NPR effects along certain directions. Another practical problem with those structures is the limitation of manufacturing processes due to the complex structures and deformation mechanism of the geometry models [34]. This is why the study is still at theoretical stage. However, with the development of modern manufacturing methods, additive (e.g. layered) manufacturing techniques provide a possibility to fabricate NPR complex structures. A few efforts have been done to fabricate 2D NPR structures using layered manufacturing techniques [19], [44], but the work is still limited on periodic structures which only have in-plane NPR effects along certain directions. Although the structures are built in 3D space, they can hardly be considered as 3D structural materials. So far, the only 3D metamaterial with a negative Poisson’s ratio was developed by Buckmann et al. [45] using dip-in direct laser writing optical lithography. Still, their effort cannot breakthrough the limitations of the orientation-dependent properties. In spite of the limitations of the previous studies, those pioneering works have demonstrated that the development of modern manufacturing techniques can provide great possibilities in the research of new NPR materials. Actually, the latest layered manufacturing techniques could fabricate a material using different materials at different dimensional scales. That makes possible to involve an embedding structure with intricate geometries into a host matrix to form a new composite structure.

In this paper, a 2D composite structure, which is formed by embedding re-entrant inclusions into a matrix with random locations and orientations, is developed based on a numerical study. The effects of random re-entrant unit cells on the deformation mechanism of the overall material are analyzed using the finite-element method. The results show that the developed composite structure has an isotropic NPR effect. Although the study is carried out in 2D, the similar concepts can easily be extended to 3D using re-entrant tetrahedron inclusions.