From single-scale turbulence models to multiple-scale and subgrid-scale models by Fourier transform

A theoretical method based on mathematical physics formalism that allows transposition of turbulence modeling methods from URANS (unsteady Reynolds averaged Navier–Stokes) models, to multiple-scale models and large eddy simulations (LES) is presented. The method is based on the spectral Fourier transform of the dynamic equation of the two-point fluctuating velocity correlations with an extension to the case of non-homogenous turbulence. The resulting equation describes the evolution of the spectral velocity correlation tensor in wave vector space. Then, we show that the full wave number integration of the spectral equation allows one to recover usual one-point statistical closure whereas the partial integration based on spectrum splitting gives rise to partial integrated transport models (PITM). This latter approach, depending on the type of spectral partitioning used, can yield either a statistical multiple-scale model or subfilter transport models used in LES or hybrid methods, providing some appropriate approximations are made. Closure hypotheses underlying these models are then discussed by reference to physical considerations with emphasis on identification of tensorial fluxes that represent turbulent energy transfer or dissipation. Some experiments such as the homogeneous axisymmetric contraction, the decay of isotropic turbulence, the pulsed turbulent channel flow and a wall injection induced flow are then considered as typical possible applications for illustrating the potentials of these models.      

How to compute plastic zones of heterogeneous materials: A simple approach using classical continuum and fracture mechanics

This is a short technical paper on how to use classical continuum and fracture mechanics to calculate the plastic zones caused by cracks on heterogeneous or composite materials. As an example, a sample consisting of an α-phase and β-phase is used. A crack is introduced to the sample, and stress is then applied. The plastic zone in front of the crack resulting from the applied stress is then calculated using commercial software. The concept uses two-level modeling: a global model using homogenized stiffness from a unit cell of heterogeneous material and a local model for the α-phase and β-phase. While this paper is written for general purposes, a concrete example using ferrite and martensite is also presented along with the experimental data. General agreement between the model and the experiment is observed. This method eliminates the need for a cumbersome analytical approach.     

Trans-scale mechanics： looking for the missing links between continuum and micro/nanoscopic reality

Problems involving coupled multiple space and time scales offer a real challenge for conventional frame-works of either particle or continuum mechanics. In this paper, four cases studies （shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics） are provided to illustrate the three levels of trans-scale problems （problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/ molecules and a finite size body at micro/nano-level） and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.     

Review of terramechanics models and their applicability to real-time applications

ISTVS embarked on a project in 2016 that aims at updating the current ISTVS standards related to nomenclature, definitions, and measurement techniques for modelling, parameterizing, and, respectively, testing and validation of soft soil parameters and vehicle running gear-terrain interaction. As part of this project, a comprehensive literature review was conducted on the parameterization of fundamental terramechanics models. Soil parameters of the empirical models to assess off-road vehicle mobility, and parameters of the models to characterize the response of the terrain interacting with running gears or plates from the existing terramechanics literature and other researchers’ reports were identified. This review documents and summarizes the modelling approaches that may be applicable to real-time applications of terramechanics in simulation, as well as in controller design.     

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.     

Multi-mechanism models: Present state and future trends

The paper gives a review of the main features introduced in the multi-mechanism models, of the present possibilities and of further developments.In the two last decades, various materials and mechanical effects were studied using multi-mechanism model types. Particular attention was given to the possible link between the mechanism definitions and the physical deformation sources. The main results of these works are first recalled. Propositions of future development for the multi-mechanism models are finally given.     

On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models

Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model.In the interest of simplicity the microscopic model is a discrete periodic nonlinear elastic structure. The corresponding macroscopic model derived from it is a continuum model involving higher order gradients in the displacements. Attention is focused on the simplest such model, namely the one whose energy density involves only the second order gradient of the displacement. The discrete to continuum comparisons are done for a boundary value problem involving two different types of macroscopic material behavior. In addition the issues of stability and imperfection sensitivity of the solutions are also investigated.     

Dislocation dynamics: from microscopic models to macroscopic crystal plasticity

In this paper we study the connection between four models describing dislocation dynamics: a generalized 2D Frenkel-Kontorova model at the atomic level, the Peierls-Nabarro model, the discrete dislocation dynamics and a macroscopic model with dislocation densities. We show how each model can be deduced from the previous one at a smaller scale.      

Stabilized interface methods for mechanical joints: Physics-based models and variationally consistent embedding

This paper presents the application of a new method for interfacial modeling utilizing a merger of continuous Galerkin and discontinuous Galerkin concepts to simulate the behavior of mechanical joints. The interfacial flux terms arising naturally from the discontinuous Galerkin treatment provide a mechanism to embed friction models in a variationally consistent fashion. Due to the unbiased implementation of the interface, facilitated by avoiding the master–slave concept, the deformation of the two interacting surfaces conforms to the local material and geometric attributes of the surfaces. This results in a better preservation of physics in interface mechanics. Additionally, the method is incorporated into a Variational Multiscale framework that comes equipped with a built-in error estimation module, providing numerical estimation of convergence and distinguishing discretization errors from modeling errors. A series of quasi-static numerical simulations of a lap joint under fretting conditions are conducted to compare the performance of two friction models: (i) classical Coulomb friction model and (ii) physics-based multiscale model. Hysteresis study of a three-dimensional double-bolted lap joint for the two friction models is also presented and the computed results are shown to be consistent between conforming and nonconforming meshes.     

Shocks and slip systems: Predictions from a mesoscale theory of continuum dislocation dynamics

Exploring a recently developed mesoscale continuum theory of dislocation dynamics, we derive three predictions about plasticity and grain boundary formation in crystals. (1) There is a residual stress jump across grain boundaries and plasticity-induced cell walls as they form, which self-consistently acts to attract neighboring dislocations; residual stress in this theory appears as a remnant of the driving force behind wall formation under both polygonization and plastic deformation. We derive the predicted asymptotic late-time dynamics of the grain-boundary formation process. (2) During grain boundary formation at high temperatures, there is a predicted cusp in the elastic energy density. (3) In early stages of plasticity, when only one type of dislocation is active (single-slip), cell walls do not form in the theory; instead we predict the formation of a hitherto unrecognized jump singularity in the dislocation density.     

Linking discrete particle simulation to continuum process modelling for granular matter: Theory and application

Two approaches are widely used to describe particle systems: the continuum approach at macroscopic scale and the discrete approach at particle scale. Each has its own advantages and disadvantages in the modelling of particle systems. It is of paramount significance to develop a theory to overcome the disadvantages of the two approaches. Averaging method to link the discrete to continuum approach is a potential technique to develop such a theory. This paper introduces an averaging method, including the theory and its application to the particle flow in a hopper and the particle-fluid flow in an ironmaking blast furnace.     

Modulation of solitary waves and formation of stable attractors in granular scalar models subjected to on-site perturbation

Present work concerns the propagation of solitary waves in the array of coupled, uncompressed granular chains subjected to onsite perturbation. We devise a special analytical procedure depicting the modulation of solitary pulses caused by the inter-chain interaction as well as by the on-site perturbations of a general type. The proposed analytical procedure is very efficient in depicting both the transient response characterized by significant energy fluctuations between the chains as well as in predicting the formation of stable attractors corresponding to a steady state response. We confirm the validity of a general analytical procedure with several specific setups of granular scalar models. In particular we consider the response of the array of coupled granular chains free of perturbation as well as the arrays subject to the basic type of on-site perturbations such as the ones induced by the uniform and random elastic foundation, dissipation. Additional interesting finding made in the present study corresponds to the granular arrays subject to a special type of on-site perturbation containing the terms leading to the two opposing effects namely dissipation and energy sourcing. Interestingly enough this type of perturbation may lead to the formation of stable attractors. By the term attractors we refer to the stable, stationary pulses simultaneously forming on all the coupled chains and propagating with the same phase speed. It is important to emphasize that the analytical procedure developed in the first part of the study predicts the formation of stable attractors through a typical saddle–node bifurcation. Moreover, results of the reduced model are found to be in a spectacular agreement with those of the direct numerical simulations of the true model.     

Multi-scale and multi-order singularity approach to non-equilibrium mechanics: Coupling of atomic-micro-macro damage

Offered in this work is the development of a macro/meso/micro model that covers the lineal scale of 10⁻¹¹ to 10⁰ by application of the volume energy density function. Boundary constraints and defect geometries are shown to play a role at the smaller scale in the same way as those at the macroscopic scale. Different orders of stress (or energy density) singularities are used to describe the defect geometry and prevailing constraint via the boundary conditions in a way similar to singularity adopted in classical fracture mechanics. Two classes of singularities have been identified in addition to classical one without violating the finiteness conditions of the local displacement and energy density. Still the connection of results from the different scales is no small task and is made possible by application of a scale multiplier. It is determined by considering the interactive effects of the parameters at the different scales from the atomic to the macroscopic. Unlike the classical boundary value problem approach, application of the scale multiplier has led to closed-form asymptotic multiscale solutions that otherwise would not have been made possible. The procedure is demonstrated for the anti-plane shear of a macro-micro-atomic model that accounts for imperfection at the different scales Published in Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 3–22, January 2006.     

Approaches in computational welding mechanics applied to additive manufacturing: Review and outlook

The development of computational welding mechanics (CWM) began more than four decades ago. The approach focuses on the region outside the molten pool and is used to simulate the thermo-metallurgical-mechanical behaviour of welded components. It was applied to additive manufacturing (AM) processes when they were known as weld repair and metal deposition. The interest in the CWM approach applied to AM has increased considerably, and there are new challenges in this context regarding welding. The current state and need for developments from the perspective of the authors are summarised in this study.     

Contradiction between the C‐property and mass conservation in adaptive grid based shallow flow models: cause and solution

When performing shallow flow simulations on adaptive grids, the C‐property (i.e. conservation property) and the mass conservation may not be simultaneously preserved, that is, either C‐property or mass conservation is likely to be violated following grid refining or coarsening. The cause of such a contradiction is analyzed in detail in this work, which essentially links to the reconstruction of bed and flow information in those newly created cells during grid adaptation. An effective approach is subsequently proposed to resolve the contradiction by locally modifying the bed elevation in certain problematic cells when reconstructing flow information using linear interpolation as part of the grid adaptation procedure. Copyright © 2015 John Wiley & Sons, Ltd.