Random packings of granular chains are presented as a model polymer system to investigate the contribution of entanglements to strain-stiffening in the absence of Brownian motion. The chain packings are sheared in triaxial compression experiments. For short chain lengths, these packings yield when the shear stress exceeds a the scale of the confining pressure, similar to packings of spherical particles. In contrast, packings of chains which are long enough to form loops exhibit strain-stiffening, in which the effective stiffness of the material increases with strain, similar to many polymer materials. The latter packings can sustain stresses orders-of-magnitude greater than the confining pressure, and do not yield until the chain links break. X-ray tomography measurements reveal that the strain-stiffening packings contain system-spanning clusters of entangled chains.PACS numbers: 83.80. Fg, 81.70.Tx, 62.20.mm, 61.41.+e Most materials, ranging from crystalline metals to piles of granular material, become weaker the further they are strained. On the other hand, many polymeric materials are known to strain-stiffen, in which the effective stiffness of the material increases as the material is strained further [1]. Theories suggest strain-stiffening could depend on many factors including chain stiffness, density, temperature, strain rate, and in particular on structures such as entanglements between different chains [2][3][4][5]. However, it has not been possible to directly measure entanglements in experiments because in polymers these structures occur on very small scales. In this letter we present a new experimental approach to investigate the role of entanglements in strain-stiffening using a model system of granular chains consisting of millimeter-scale beads connected by flexible links. Such a macroscopic system has advantages over molecular polymer systems for investigating entanglement. First, the macroscopic size allows for imaging to measure the precise positions of each particle and link in the structure. Second, we can isolate entanglement effects from temperature and strain rate dependent effects because the macroscopic chains have no inherent time scales due to Brownian motion or relaxation.Granular chains have been considered as model polymers in previous studies to characterize the packing structure around the jamming transition for chain packings [6][7][8]. Such chains form tight loops when packed which defines a characteristic loop size and persistence length in analogy to polymers. The free volume in the packing was found to increase with and level off when the chain length exceeded the persistence length, analogous to the behavior of the glass transition temperature for polymers [7]. Similarly, tumbling strings have been considered as a macroscopic model polymer system, in which knotting was found to exhibit statistics analogous to thermodynamic systems [9]. While these works characterized the structures of macroscopic model polymer systems, the stress response of such systems have not previous...

We perform a hybrid experimental and numerical study of the localization of deformation in thin spherical elastic shells under indentation. Past a critical indentation, the deformation of the shell ceases to be axisymmetric and sharp points of localized curvature form. In plates, these sharp points are known as d-cones. By way of analogy, regions of localization in shells are referred to as s-cones, for 'shell-cones'. We quantify how the formation and evolution of s-cones is affected by the indenter's curvature. Juxtaposing results from precision model experiments and Finite Element simulations enables the exploration of the frictional nature of the shell-indenter contact. The numerics also allow for a characterization of the relative properties of strain energy focusing, at the different loci of localization. The predictive power of the numerics is taken advantage of to further explore parameter space and perform numerical experiments that are not easily conducted physically. This combined experimental and computational approach allows us to gain invaluable physical insight towards rationalizing this geometrically nonlinear process.

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We present results from a numerical investigation of the localization of deformation in thin elastomeric spherical shells loaded by differently shaped indenters. Beyond a critical indentation, the deformation of the shell ceases to be axisymmetric and sharp structures of localized curvature form, referred to as "s-cones," for "shell-cones." We perform a series of numerical experiments to systematically explore the parameter space. We find that the localization process is independent of the radius of the shell. The ratio of the radius of the shell to its thickness, however, is an important parameter in the localization process. Throughout, we find that the maximum principal strains remain below 6%, even at the s-cones. As a result, using either a linear elastic (LE) or hyperelastic constitutive description yields nearly indistinguishable results. Friction between the indenter and the shell is also shown to play an important role in localization. Tuning this frictional contact can suppress localization and increase the load-bearing capacity of the shell under indentation.

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