Realistic Numerical Analysis of a Bioinspired Layered Composite with a Crack: Robust Scaling Laws and Crack Arrest

Abstract

Nacre is a prototype of natural strong and tough materials and has been studied extensively. However, no numerical models have been developed that faithfully reflect the layered structure made of alternately stacked soft and hard layers, in order to study the fracture toughness in the presence of a macroscopic crack. In this study, we construct a realistic numerical model by finite element method (FEM) that explicitly takes the layered structure into account and study the stress and deformation fields around a crack. Although we reflect a realistic layered structure, we remarkably find simple scaling laws for the elastic fields, which predict considerable reduction of the stress concentration around the crack tips in exchange for enhancement of the deformation. We also find that the divergence at the crack tips in the scaling law for the stress field is cut-off at the scale of layers, which significantly restrains the maximum stress appearing at the tips. We further find that a crack tip is necessarily stopped at a soft layer and this crack arrest guarantees strong effects of the reduction of the stress concentration and cut-off of the stress singularity. In addition, these toughening mechanisms lead to the prediction of the correct order of the fracture energy experimentally reported in a seminal paper. These mechanisms and the FEM model developed here will be useful for the development of artificial tough advanced materials.Tough and strong materials in nature quite often exhibit magnificent hierarchical structures. Nacre is probably the most well studied material as such; it is a shining layered composite found inside certain sea shells or on the surface of pearl. This material is remarkable in that only a small amount of the soft component between the layers of the hard component makes the fracture energy a few thousand times as high as that of the monolithic material made of the hard component. [1,2] Accordingly various toughening mechanism of nacre have been proposed, [3][4][5][6][7][8] and nacre has bioinspired a number of remarkable materials. [9][10][11][12][13] Here, in particular, we focus on a simple model of nacre [14] that provides simple understandings of the type recently revealed for spider webs, [15][16][17] (although it does not include recently found complex structures beyond the layered structure [18,19] ). This is because simple scaling laws were obtained for the model to predict the correct order of the fracture energy of nacre, on the basis of analytical solutions for two crack problems. [14,20] However, in this model the layered structure is coarsegrained or homogenized: the continuum model is valid on scales larger than the layer period and thus can predict nothing on smaller scales. In particular, this model is insensitive to the position of crack tips, i.e. whether the tips are located in soft or hard layers. To study such issues, we need to construct appropriate numerical models that allow us to seek what happens at scales inaccessible by the coarsegrained continuum model. One such ex...

Simple Model for the Toughness of a Helical Structure Inspired by the Exoskeleton of Lobsters

“…In fact, as suggested in Ref. 33, this result is robust (i.e., it is applicable in a wider range than theoretically expected).…”

“…Note that the results for nacre have recently been suggested to be robust. 33) From the expression for the fracture energy for the exoskeleton, it was clarified that the helical structure and the combination of hard and soft elements cooperate together to make the composite tougher than a nonhelical composite in any directions against a crack larger than the helical pitch. Furthermore, it was clarified how the simple model of the exoskeleton is optimized in terms of toughness by adjusting the volume fractions of the bundles and the calcium carbonate matrix and by preparing two cuticles, an exocuticle and endocuticle, with different helical pitches.…”

“…The present results are considered to be robust because the scaling relations for the nacre model have recently been shown to be robust through the finite-element method (FEM). 33) On the basis of our results obtained from the simple model, we can answer the following questions purely from physical viewpoints: (1) Why does the cuticle in the exoskeleton form a helical structure? (2) Why is the cuticle composed of soft and hard components, i.e., bundles and a matrix?…”

“…In addition, we consider below only the leading contribution in the limit of the small parameter " (see below), and we do not integrate the detailed (sub-)structures beyond the bundle and matrix structure, such as the canal structure, into our theory. 22,23) This level of simplification still merits consideration for the following reasons: (1) Frequently, scaling relations are robust, i.e., they are practically valid in a wider range of values than theoretically expected, as seen in various examples such as the FEM calculations for nacre 33) and a number of studies on wetting (for example, see Ref. 34).…”

Modeling the Mechanism of Inelastic Shear at Fiber Interfaces in Material with Unidirectional Structure by Thin Perfectly Plastic Interlayer

“…3, the inelastic shear surface that intersects the crack near its tip significantly reduces the SIF even at high yield strength of the interlayer. Thus, the low SIF is typical not only for the crack with the tip within the soft layer between the load-bearing elements in the fiber (layered) structure [23,24]. The results in Fig.…”