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We revisit the classic stability problem of the buckling of an inextensible, axially compressed beam on a nonlinear elastic foundation with a semi-analytical approach to understand how spatially localized deformation solutions emerge in many applications in mechanics. Instead of a numerical search for such solutions using arbitrary imperfections, we propose a systematic search using branch-following and bifurcation techniques along with group-theoretic methods to find all the bifurcated solution orbits (primary, secondary, etc.) of the system and to examine their stability and hence their observability. Unlike previously proposed methods that use multi-scale perturbation techniques near the critical load, we show that to obtain a spatially localized deformation equilibrium path for the perfect structure, one has to consider the secondary bifurcating path with the longest wavelength and follow it far away from the critical load. The novel use of group-theoretic methods here illustrates a general methodology for the systematic analysis of structures with a high degree of symmetry.

Nanostructures are technological devices constructed on a nanometer length scale more than a thousand times thinner than a human hair. Due to the unique properties of matter at this scale, such devices offer great potential for creating novel materials and behaviors that can be leveraged to benefit mankind. This paper addresses a particular challenge involved in the design of nanostructures—their stochastic or apparently random response to external loading. This is because fundamentally the function that relates the energy of a nanostructure to the arrangement of its atoms is extremely nonconvex, with each minimum corresponding to a possible equilibrium state that may be visited as the system responds to loading. Traditional atomistic simulation techniques are not capable of systematically addressing this complexity. Instead, we construct an equilibrium map (EM) for the nanostructure, analogous to a phase diagram for bulk materials, which fully characterizes its response. Using the EM, definitive predictions can be made in limiting cases and the spectrum of responses at any desired loading rate can be obtained. The latter is important because standard atomistic methods are fundamentally limited, by computational feasibility, to simulations of loading rates that are many orders of magnitude faster than reality. In contrast, the EM-based approach makes possible the direct simulation of nanostructure experiments. We demonstrate the method’s capabilities and its surprisingly complex results for the case of a nanoslab of nickel under compression.

Continuum mechanics and thermodynamics are foundational theories of many fields of science and engineering. This book presents a fresh perspective on these fundamental topics, connecting micro- and nanoscopic theories and emphasizing topics relevant to understanding solid-state thermo-mechanical behavior. Providing clear, in-depth coverage, the book gives a self-contained treatment of topics directly related to nonlinear materials modeling. It starts with vectors and tensors, finite deformation kinematics, the fundamental balance and conservation laws, and classical thermodynamics. It then discusses the principles of constitutive theory and examples of constitutive models, presents a foundational treatment of energy principles and stability theory, and concludes with example closed-form solutions and the essentials of finite elements. Together with its companion book, Modeling Materials, (Cambridge University Press, 2011), this work presents the fundamentals of multiscale materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering.

The unusual properties of shape memory alloys (SMAs) result from a lattice level martensitic transformation (MT) corresponding to an instability of the SMAs crystal structure. Currently, there exists a shortage of material models that can capture the details of lattice level MTs occurring in SMAs and that can be used for efficient computational investigations of the interaction between MTs and larger-scale features found in typical materials. These larger-scale features could include precipitates, dislocation networks, voids, and even cracks. In this article, one such model is developed for the SMA AuCd. The model is based on effective interaction potentials (EIPs). These are atomic interaction potentials that are explicit functions of temperature. In particular, the Morse pair potential is used and its adjustable coefficients are taken to be temperature dependent. An extensive exploration of the Morse pair potential is performed to identify an appropriate functional form for the temperature dependence of the potential parameters. A fitting procedure is developed for the EIPs that matches, at suitable temperatures, the stress-free equilibrium lattice parameters, instantaneous bulk moduli, cohesive energies, thermal expansion coefficients, and heat capacities of FCC Au, HCP Cd, and the B2 cubic austenite phase of the Au-47.5at%Cd alloy. The resulting model is explored using branch-following and bifurcation techniques. A hysteretic temperature-induced MT between the B2 cubic and B19 orthorhombic crystal structures is predicted. This is the behavior that is observed in the real material. In addition to reproducing the important properties mentioned above, the model predicts, to reasonable accuracy, the transformation strain tensor and captures the latent heat and thermal hysteresis to within an order of magnitude.

Solid-to-solid martensitic phase transformations are responsible for the remarkable behavior of shape memory alloys. There is currently a need for shape memory alloys with improved corrosion, fatigue, and other properties. The development of new accurate models of martensitic phase transformations based on the material’s atomic composition and crystal structure would lead to the ability to computationally discover new improved shape memory alloys. This paper explores the Effective Interaction Potential method for modeling the material behavior of shape memory alloys. In particular, an extensive parameter study of the Morse pair potential model of the stress-free B2 cubic crystal is performed. Results for the stability, potential energy, current unit cell volume, instantaneous bulk modulus, and the two instantaneous cubic shear moduli are presented and discussed. It is found that an Effective Interaction Potential model based on the Morse potential is appropriate for modeling transformations between the B2 cubic structure and the B19 orthorhombic structure, but is not likely to be capable of simulating the B2 cubic to B19′ monoclinic transformation found in the popular shape memory alloy NiTi. In fact, this conclusion may be extended to all types of pair interaction potential models.

Interatomic potentials (IPs) are reduced-order models for calculating the potential energy of a system of atoms given their positions in space and species. IPs treat atoms as classical particles without explicitly modeling electrons and thus are computationally far less expensive than first-principles methods, enabling molecular simulations of significantly larger systems over longer times. Developing an IP is a complex iterative process involving multiple steps: assembling a training set, designing a functional form, optimizing the function parameters, testing model quality, and deployment to molecular simulation packages. This paper introduces the KIM-based learning-integrated fitting framework (KLIFF), a package that facilitates the entire IP development process. KLIFF supports both physics-based and machine learning IPs. It adopts a modular approach whereby various components in the fitting process, such as atomic environment descriptors, functional forms, loss functions, optimizers, quality analyzers, and so on, work seamlessly with each other. This provides a flexible framework for the rapid design of new IP forms. Trained IPs are compatible with the Knowledgebase of Interatomic Models (KIM) application programming interface (API) and can be readily used in major materials simulation packages compatible with KIM, including ASE, DL_POLY, GULP, LAMMPS, and QC. KLIFF is written in Python with computationally intensive components implemented in C++. It is parallelized over data and supports both shared-memory multicore desktop machines and high-performance distributed memory computing clusters. We demonstrate the use of KLIFF by fitting a physics-based Stillinger--Weber potential and a machine learning neural network potential for silicon. The KLIFF package, together with its documentation, is publicly available at: https://github.com/openkim/kliff.

We present a systematic methodology, built within the Open Knowledgebase of Interatomic Models (OpenKIM) framework (https://openkim.org), for quantifying properties of grain boundaries (GBs) for arbitrary interatomic potentials (IPs), GB character, and lattice structure and species. The framework currently generates results for symmetric tilt GBs in cubic materials, but can be readily extended to other types of boundaries. In this paper, GB energy data are presented that were generated automatically for Al, Ni, Cu, Fe, and Mo with 225 IPs; the system is installed on openkim.org and will continue to generate results for all new IPs uploaded to OpenKIM. The results from the atomistic calculations are compared to the lattice matching model, which is a semi-analytic geometric model for approximating GB energy. It is determined that the energy predicted by all IPs (that are stable for the given boundary type) correlate closely with the energy from the model, up to a multiplicative factor. It thus is concluded that the qualitative form of the GB energy versus tilt angle is dominated more by geometry than the choice of IP, but that the IP can strongly affect the energy level. The spread in GB energy predictions across the ensemble of IPs in OpenKIM provides a measure of uncertainty for GB energy predictions by classical IPs.