We present a new type of a truly three-dimensional cubic negative stiffness lattice structure that can achieve energy absorption and recover its original configuration under cyclic loading in excess of a strain of approximately 20%. This structure was composed of multiple unit cells exploiting negative stiffness from controlled elastic buckling. Structural properties were designed to be tuned by adjusting geometry. The effective Poisson’s ratio was equal to zero regardless of the constituent material. Geometric parameters that can lead to the desired energy absorption without yielding the structure were determined by finite element analysis. We then fabricated samples representing the lattice structure with different sizes through additive manufacturing and performed cyclic loading experiments to capture stress-strain hysteresis loops. Results clearly showed that the designed structure was capable of absorbing mechanical energy effectively with a full recovery of geometry in three principal directions and that the amount of energy absorbed during cyclic loading increases with its size.

In this study, an energy absorption lattice, comprised of multiple tetra-beam-plate unit cells with negative stiffness, was designed, fabricated by selective laser sintering method, and analyzed both numerically and experimentally. Snap-through behavior of the unit cell developed due to negative stiffness caused by geometric nonlinearity from large deflection of the constituent elastic beams, resulting in energy absorption. A criterion for the unit cell to achieve the snap-through behavior was investigated numerically in terms of the beam slenderness ratio and the inclined angle. This approach was chosen to facilitate control of energy dissipation performance and further design space such as tuning force threshold. The unit cell with the selected geometric parameters was then created and used to construct the energy absorption lattice. Load-displacement relationships of the lattices obtained from cyclic loading tests disclosed an area enclosed by two distinct loading and unloading curves, which indicates energy dissipation. This was shown both numerically and experimentally. Drop tests were also performed to investigate energy loss of the lattices due to an impact. An energy absorption phenomenon was revealed by observing a reduced rebound height when the lattice exhibited the snap-through behavior.

Chiral three-dimensional cubic lattices are developed with rigid cubical nodules and analyzed via finite element analysis. The lattices exhibit geometry dependent Poisson’s ratio that can be tuned to negative values. Poisson’s ratio tends to zero as the cubes become further apart. The lattices exhibit stretch–twist coupling. Such coupling cannot occur in a classical elastic continuum but it can occur in a chiral Cosserat solid.

Chiral three‐dimensional isotropic cubic lattices with rigid cubical nodules and multiple deformable ribs are developed and analyzed via finite element analysis. The lattices exhibit geometry‐dependent Poisson's ratio that can be tuned to negative values. Poisson's ratio decreases from positive to negative values as the number of cells increases. Isotropy is obtained by adjustment of aspect ratio. The lattices exhibit significant size effects. Such a phenomenon cannot occur in a classical elastic continuum but it can occur in a Cosserat solid.

Thermoelastic triangular cell lattices composed of bi-material curved ribs were designed and analyzed by finite element simulation. Positive, negative, or zero thermal expansion was possible by varying rib curvature if joints can pivot freely, as expected. Welded or bonded joints result in nonzero expansion but smaller in magnitude than that of a constituent material having higher thermal expansion coefficient. The effects of rib curvature variation for bonded joints were found to be negligible. Rib slenderness for both joints did not influence the coefficient of thermal expansion. We present a square lattice with bonded joints that has zero net thermal expansion; each curved bi-material rib has zero expansion.

We perform a micromechanical analysis of general isotropic non-cohesive particulate materials idealized as three-dimensional random assemblies of uniform spheres with a simple linear elastic inter-particle contact force law and inter-particle Coulomb friction law. We obtain analytical relationships between the inter-particle friction coefficient \(\mu\) (or inter-particle friction angle \(\phi _\mu = \tan ^{-1} \mu\)) on the microscale and the material friction angle \(\phi\) on the macroscale. Our micromechanical analysis directly employs force and moment equilibrium (together with compatibility and the contact constitutive assumptions noted) rather than energy methods, and thus can account for the effects of particle rotation, and in particular the effects of mechanisms or zero-energy modes due to particle rotation. To explore the effects of particle rotation, we perform analyses with particle rotation either allowed or prohibited. To validate the analytical results obtained here, we compare the \(\phi\) versus \(\phi _\mu\) curves determined theoretically to those obtained by the discrete element method (DEM) for six randomly packed specimens of 3430–29, 660 uniform spherical elements with uniform inter-element Coulomb friction in Fleischmann et al. in Geotech Geol Eng 32(4):1081–1100, (2014). The \(\phi\) versus \(\phi _\mu\) curves derived here show remarkable agreement with those obtained via DEM simulations in Fleischmann et al. in Geotech Geol Eng 32(4):1081–1100, (2014), especially for the case in which particle rotation is not artificially restrained.

Lattices of controlled thermal expansion are presented based on planar chiral lattice structure with Poisson's ratio approaching -1. Thermal expansion values can be arbitrarily large positive or negative. A lattice was fabricated from bimetallic strips and the properties analyzed and studied experimentally. The effective thermal expansion coefficient of the lattice is about =-3.5x10-4K-1. This is much larger in magnitude than that of constituent metals. Nodes were observed to rotate as temperature was changed corresponding to a Cosserat thermoelastic solid. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We perform numerical simulations using the discrete element method (DEM) to determine yield surfaces for large samples of randomly packed uniform spheres with constant normal and tangential contact stiffnesses (linear spring model) and uniform inter-particle friction coefficient μ , for a large range of values of the inter-particle friction coefficient μ . The beauty of DEM is that the micromechanical properties of the spheres, especially the inter-particle friction coefficient μ , are known exactly. Further, simulations can be performed with particle rotation either prohibited or unrestrained, which provides an effective means for evaluating analytical models that employ these assumptions. We compare the resulting yield surfaces to the Mohr–Coulomb, Matsuoka–Nakai, Lade–Duncan, and Drucker–Prager yield surfaces, and determine the relationship between the resulting material friction angle ϕ on the macroscale and the inter-particle friction coefficient μ (or the inter-particle friction angle ϕ μ ) on the microscale. We find the Lade–Duncan yield surface provides the best agreement, by far, with the simulations in all cases. We also monitor inter-particle friction work and particle rotation within each specimen during the DEM simulations, both globally and on a particle-by-particle basis, and we compare the results obtained from DEM simulations in which the spheres were allowed full three-dimensional translational and rotational freedom of motion and DEM simulations in which particle rotation was prohibited.

Plesha, Gray, & Costanzo's Engineering Mechanics, 2e is the Problem Solver's Approach for Tomorrow's Engineers. Based upon a great deal of classroom teaching experience, Plesha, Gray, & Costanzo provide a visually appealing learning framework to your students. The look of the presentation is modern, like the other books the students have experienced, and the presentation itself is relevant, with examples and exercises drawn from the world around us, not the world of sixty years ago. Examples are broken down in a consistent manner that promotes students' ability to setup a problem and easily solve problems of incrementally harder difficulty. Engineering Mechanics is also accompanied by McGraw-Hill's Connect which allows the professor to assign homework, quizzes, and tests easily and automatically grades and records the scores of the students' work. Most problems in Connect are randomized to prevent sharing of answers and most also have a multi-step solution which helps move the students' learning along if they experience difficulty. Engineering Mechanics, 2e by Plesha, Gray, & Costanzo, a new dawn for statics and dynamics.

Results obtained from numerical simulations of direct (or ring) shear tests on ASTM standard graded (Ottawa) sand using the discrete-element method with periodic boundary conditions in both two and three dimensions are reported. The data obtained from these simulations are quantitatively compared to experimental data for ASTM standard graded sand. The results show that the three-dimensional (3D) effects of nonplanar interparticle contact forces and particle motion are significant in the 3D discrete-element method simulations, even during nominally two-dimensional (2D) shear flow. Moreover, the 3D discrete-element method simulations accurately predict the peak and residual friction angles of ASTM standard graded sand. On the other hand, the 2D discrete-element method simulations fail to accurately predict the peak and residual friction angles. It is argued that the failure of the 2D discrete-element method simulations and the success of the 3D discrete-element method simulations in providing quantitatively accurate predictions of peak and residual friction angles are attributable largely to the respective absence or presence of 3D effects, including nonplanar interparticle contact forces and nonplanar particle motion, in these discrete-element method simulations. (C) 2013 American Society of Civil Engineers.

A numerical finite element contact model is developed to make use of the high precision surface topography data obtained at the nanoscale by atomic force microscopy or other imaging techniques while minimizing computational complexity. The model uses degrees of freedom that are normal to the surface, and uses the Boussinesq solution to relate the normal load to the long-range surface displacement response. The model for contact between two rough surfaces is developed in a step-by-step manner, taking into account the far-field effects of the loads developed at asperities that have come to contact in previous steps. Method accuracy is verified by comparison to simple test cases with well-defined analytical solutions. Agreement was found to be within 1 % for a wide range of practical loads for the high precision models. Applicability of extrapolation from lower precision models is presented. The real contact area estimates for micrometer-size tribology test machine surfaces are calculated and convergence behavior with mesh refinement is investigated.

In Part I, Fleischmann et al. (2013), we performed theoretical analyses of three cubic packings of uniform spheres (simple, body-centered, and face-centered) assuming no particle rotation, employed these results to derive the effective elastic moduli for a statistically isotropic particulate material, and assessed these results by performing numerical discrete element method (DEM) simulations with particle rotations prohibited. In this second part, we explore the effect that particle rotation has on the overall elastic moduli of a statistically isotropic particulate material. We do this both theoretically, by re-analyzing the elementary cells of the three cubic packings with particle rotation allowed, which leads to the introduction of an internal parameter to measure zero-energy rotations at the local level, and numerically via DEM simulations in which particle rotation is unrestrained. We find that the effects of particle rotation cannot be neglected. For unrestrained particle rotation, we find that the self-consistent homogenization assumption applied to the locally body-centered cubic packing incorporating particle rotation effects most accurately predicts the measured values of the overall elastic moduli obtained from the DEM simulations, in particular Poisson's ratio. Our new self-consistent results and theoretical modeling of particle rotation effects together lead to significantly better theoretical predictions of Poisson's ratio than all prior published results. Moreover, our results are based on a direct micromechanics analysis of specific geometrical packings of uniform spheres, in contrast to prior theoretical analyses based on hypotheses involving overall inter-particle contact distributions. Thus, our results permit a direct assessment of the reasons for the theory–experiment discrepancies noted in the literature with regard to previous theoretical derivations of the macroscopic elastic moduli for particulate materials, and our new theoretical results greatly narrow such discrepancies.

Notation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.

The discrete element method (DEM) is developed in this study as a general and robust technique for unified two‐dimensional modelling of the mechanical behaviour of solid and particulate materials, including the transition from solid phase to particulate phase. Inter‐element parameters (contact stiffnesses and failure criteria) are theoretically established as functions of element size and commonly accepted material parameters including Young's modulus, Poisson's ratio, ultimate tensile strength, and fracture toughness. A main feature of such an approach is that it promises to provide convergence with refinement of a DEM discretization. Regarding contact failure, an energy criterion based on the material's ultimate tensile strength and fracture toughness is developed to limit the maximum contact forces and inter‐element relative displacement. This paper also addresses the issue of numerical stability in DEM computations and provides a theoretical method for the determination of a stable time‐step. The method developed herein is validated by modelling several test problems having analytic solutions and results show that indeed convergence is obtained. Moreover, a very good agreement with the theoretical results is obtained in both elastic behaviour and fracture. An example application of the method to high‐speed penetration of a concrete beam is also given. Copyright © 2006 John Wiley & Sons, Ltd.

A model is presented to investigate contact and friction between sliding microelectromechanical systems (MEMS) surfaces. Roughness of MEMS surfaces exhibits multiscale structure. This was observed with analysis of asperities on atomic force microscope (AFM) images of real MEMS surfaces. The contact model is developed using multiple scales of surface roughness, with a single asperity contact model for the behavior of an asperity at a particular length scale, including effects such as surface forces (adhesion). The roughness information for the model is obtained from the AFM image of the MEMS surface under consideration. Results for true contact area and a prediction of the macroscopic coefficient of friction are discussed.

The Discrete Element Method (DEM) discretizes a material using rigid elements of simple shape. Each element interacts with neighboring elements through appropriate interaction laws. The number of elements is typically large and is limited by computer speed. The method has seen widespread applications to modeling particulate media and more recently to modeling solids such as concrete, ceramic, and metal. For problems with severe damage, DEM offers a number of attractive features over continuum based numerical methods, with the primary feature being a seamless transition from solid phase to particulate phase. This study illustrates the potential of DEM for modeling penetration and briefly points out its numerous advantages. A weakness of DEM is that its convergence properties are not understood. The crucial question is whether convergence is obtained as DEM element size vanishes in the limit of model refinement. The major focus of our investigation will be a careful study of convergence for modeling the degradation of a solid into fragments. Our results show that indeed convergence is obtained in several specific test problems. Moreover, elastic interelement stiffness and damping properties were proven to be particle size-independent. However, convergence in material failure due to crack growth is obtained only if the interparticle potentials are properly constructed as functions of DEM element size and bulk material properties such as elastic modulus and fracture toughness.

Investigation of contact and friction at multiple length scales is necessary for the design of surfaces in sliding microelectromechanical system (MEMS). A method is developed to investigate the geometry of asperities at different length scales. Analysis of density, height, and curvature of asperities on atomic force microscopy (AFM) images of actual silicon MEMS surfaces show these properties have a power law relationship with the sampling size used to define an asperity. This behavior and its similarity to results for fractal Weierstrass-Mandelbrot (W-M) function approximations indicate that a multiscale model is required to properly describe the surfaces.