In this paper the effect of external magnetic fields on a ferrofluid is studied numerically. The details of the proposed model are presented and compared to the literature. The governing equations are normalized and the order of magnitude of the dimensionless numbers is analyzed for realistic material parameters. Numerical difficulties arising from the multitude of parameters are discussed. As an exemplary problem a cylindrical container filled with a ferrofluid is considered. Two different magnetic excitations are investigated for different boundary conditions. First, a spherical permanent magnet is placed beneath a closed ferrofluid container and the induced motion is calculated using the finite element method and the software package FEniCS. The difference between slip and no slip boundary conditions are analyzed. Furthermore, the results are compared to a different excitation from a solenoid, which generates a uniform magnetic field inside of the ferrofluid container.

Making use of experimental data for bias extension, shearing, and point-load tests in large deformation regime for rectangular and square bi-pantographic specimens, we perform a numerical identification to fit the a priori parameters of a planar discrete spring model. The main objective of the work is to develop an automatized optimization process based on the Nelder–Mead simplex algorithm for identifying the constitutive parameters of discrete modeling of bi-pantographic structures, as well as assessing its descriptiveness and predictive capacity. The analysis allows to conclude that there exists a single set of parameters for the adopted discrete modeling such that it is descriptive and predictive for several different tests and for a wide range of deformations.

Based on the progress and advances of additive manufacturing technologies, design and production of complex structures became cheaper and therefore rather possible in the recent past. A promising example of such complex structure is a so-called pantographic structure, which can be described as a metamaterial consisting of repeated substructure. In this substructure, two planes, which consist of two arrays of beams being orthogonally aligned to each other, are interconnected by cylinders/pivots. Different inner geometries were taken into account and additively manufactured by means of fused deposition modeling technique using polyethylene terephthalate glycol (PETG) as filament material. To further understand the effect of different manufacturing parameters on the mechanical deformation behavior, three types of specimens have been investigated by means of displacement-controlled extension tests. Different slicing approaches were implemented to eliminate process-related problems. Small and large deformations are investigated separately. Furthermore, 2D digital image correlation was used to calculate strains on the outer surface of the metamaterial. Two finite-element simulations based on linear elastic isotropic model and linear elastic transverse isotropic model have been carried out for small deformations. Standardized extension tests have been performed on 3D-printed PETG according to ISO 527-2. Results obtained from finite-element method have been validated by experimental results of small deformations. These results are in good agreement with linear elastic transverse isotropic model (up to about [Formula: see text] of axial elongation), though the response of large deformations indicates a nonlinear inelastic material behavior. Nevertheless, all samples are able to withstand outer loading conditions after the first rupture, resulting in resilience against ultimate failure.

In the recent past new experimental techniques have been developed with the objective of linking generalized continuum theories with technology. So-called pantographic structures, which can be characterized as a meta-material, will be presented and investigated experimentally: Samples of different materials and dimensions are subjected to large deformation loading tests (tensile, shearing, and torsion) up to rupture, while their response to loading is recorded by an optical measurement system. 3D-digital image correlation is used to quantify the deformation.

Teeth with intact crowns rarely split or fracture, despite decades of cyclic loading and occasional unexpected overload. This is largely attributed to the presence of dentine, since cracking and fracture of enamel have been frequently reported. Dentine is similar to bone, comprising mineralised collagen fibres as a main constituent. Unlike cortical bone, however, where microcracking and damage arrest are essential for re/modelling and healing, dentine can neither remodel nor regenerate. This raises questions regarding the evolutionary benefits of toughening, leading to uncertainty whether cracks actually appear in dentine in situ. Here we study the notion that circumpulpal dentine is usually protected against, rather than damaged by severe overloads, even though it is not much more massive or stronger than it needs to be. To address this, we examined hydrated teeth still within whole jawbones of freshly-slaughtered skeletally mature pigs, mechanically loaded until fracture. Force displacement curves, optical and electron microscopy combined with 3D microstructural analysis by conventional micro-computed tomography (μCT) revealed mostly brittle fracture paths in circumpulpal crown dentine. Once overload cracks reach this mass of dentine they propagate rapidly along straight paths often parallel to the enamel flanks of the oblong shovel shaped premolars. We find infrequent signs of active toughening mechanisms with minimal crack diversion, ligament bridging and microcracking. When such toughening is seen, it mainly appears in softer dentine in the root, or near the dentine-enamel-junction (DEJ) in mantle dentine. We observed shear bands in overloaded circumpulpal dentine, due to mutual gliding of upper and lower segments. These shear bands are formed as periodic arrays of rotated dentine fragments. The 3D data consistently demonstrate the importance of the layered tooth structure, containing a stiff outer enamel shell, a soft sub-DEJ interlayer and a stiff circumpulpal dentine bulk, for deflecting cracks from splitting the tooth.

Additive manufacturing methods, commonly known as 3D printing, enable the design and manufacturing of complex and sophisticated material fabrics with a special substructure resulting in extraordinary macroscopic deformation behavior. Such a man-made structure is also referred to as a metamaterial. So called pantographic structures, which can be described as metamaterials with a substructure that is composed of two orthogonal arrays of beams connected by internal cylinders, were manufactured using fused deposition modeling technique. In order to further understand the peculiarity of its deformation behaviors, a plane sheet was also printed to be used as a comparison. Different types of experiments were performed and evaluated qualitatively by the means of digital image correlation being able to localize the initial area of out-of-plane movements in shearing tests for both specimen. Results of quasi-static standard tension and shearing tests indicate a resilient material behavior during high elastic deformations resulting in a high resistance against total failure of the structure. Furthermore, cyclic long-term tests show a viscoelastic deformation behavior of the thermoplastic material. PSs show linear as well as non-linear elastic deformation response in all experiments except the cyclic tension test.

Due to the latest advancements in 3D printing technology and rapid prototyping techniques, the production of materials with complex geometries has become more affordable than ever. Pantographic structures, because of their attractive features, both in dynamics and statics and both in elastic and inelastic deformation regimes, deserve to be thoroughly investigated with experimental and theoretical tools. Herein, experimental results relative to displacement-controlled large deformation shear loading tests of pantographic structures are reported. In particular, five differently sized samples are analyzed up to first rupture. Results show that the deformation behavior is strongly nonlinear, and the structures are capable of undergoing large elastic deformations without reaching complete failure. Finally, a cutting edge model is validated by means of these experimental results.

In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investigations. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes and the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) in exploiting the modern technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.

In the last decade, the exotic properties of pantographic metamaterials have been investigated and different mathematical models (both discrete or continuous) have been introduced. In a previous publication, a large part of the already existing literature about pantographic metamaterials has been presented. In this paper, we give some details about the next generation of research in this field. We present an organic scheme of the whole process of design, fabrication, experiments, models and image analyses.

The effect of pre-stress on the buckling behaviour of geometric unit cells of collinear square lattices is investigated experimentally and numerically. The geometric unit cells are manufactured using fused deposition modelling. Manufacturing strategies are presented which incorporate fibres subjected to pre-stress within the unit cell. The effect of pre-stressed fibres is analysed by comparing the compressive behaviour of unit cells with and without fibre reinforcement. The buckling behaviour of the unit cells is also investigated numerically by employing a parametric study within Abaqus varying the pre-stress in the fibres. The experimental test series shows that the addition of pre-stressed fibres to the system results in an increase in buckling and maximum load of 260%–480% and 220%–350% respectively. The increase strongly relates to the manufacturing quality, i.e. the bonding between the lattice material and the fibres, where a sufficient bonding yields significantly larger loads. The experimental findings on the qualitative and quantitative buckling behaviour correspond well with results obtain from the numerical study.

The development of additive manufacturing methods, such as 3D printing, allows the design of more complex architectured materials. Indeed, the main structure can be obtained by means of periodically (or quasi-periodically) arranged substructures which are properly conceived to provide unconventional deformation patterns. These kinds of materials which are ‘substructure depending’ are called metamaterials. Detailed simulations of a metamaterial is challenging but accurately possible by means of the elasticity theory. In this study, we present the steps taken for analyzing and simulating a particular type of metamaterial composed of a pantographic substructure which is periodic in space—it is simply a grid. Nevertheless, it shows an unexpected type of deformation under a uniaxial shear test. This particular behavior is investigated in this work with the aid of direct numerical simulations by using the finite element method. In other words, a detailed mesh is generated to properly describe the substructure. The metamaterial is additively manufactured using a common polymer showing nonlinear elastic deformation. Experiments are undertaken, and several hyperelastic material models are examined by using an inverse analysis. Moreover, a direct numerical simulation is repeated for all studied material models. We show that a good agreement between numerical simulations and experimental data can be attained.

In an analysis presented in 1909, Poynting (page 546 of [1]) had shown that for finite elastic deformations, the lines of greatest extension and contraction are inclined to the diagonals of the rhombus into which a square is sheared. The implication of this finding was that when slender structures are twisted, they undergo elongation as experimentally verified by Poynting's measurements with wires [1]. While many theoretical analyses have shown the possibility of Poynting and reverse (inverse) Poynting effect [2-7], measurements of such effects are rather sparse [8-11]. We present here measurements of a highly nonlinear Poynting effect, including its reversal from positive to negative (elongation to compression) direction during torsion. Such atypical behavior is exhibited by a rather exceptional material system that has a pantographic internal structure. For this material system, the classical Cauchy-type continuum model fails and a 2nd gradient continuum model is necessary to describe many of its deformation behaviors. (C) 2018 Elsevier Ltd. All rights reserved.

With the advancements in 3D printing technology, rapid manufacturing of fabric materials with complex geometries became possible. By exploiting this technique, different materials with different structures have been developed in the recent past with the objective of making generalized continuum theories useful for technological applications. So-called pantographic structures are introduced: Inextensible fibers are printed in two arrays orthogonal to each other in parallel planes. These superimposed planes are inter-connected by elastic cylinders. Five differently-sized samples were subjected to shear-like loading while their deformation response was analyzed. Results show that deformation behavior is strong non-linear for all samples. Furthermore, all samples were capable to resist considerable external shear loads without leading to complete failure of the whole structure. This extraordinary behavior makes these structures attractive to serve as an extremely tough metamaterial.