Generalized Models and Non-classical Approaches in Complex Materials 2

For ordinary people, mechanical damping is the attenuation of a motion over time under possible eventual external actions. The phenomenon is produced by the loss or dissipation of energy during motion and thus time. The concept of real time is therefore at the center of the phenomenon of damping and given the recent scientific contributions (of gravitational waves in 2016), the notion of space-time calls for reflections and comments. The systemic approach of the phenomenon taking into account the mechanical system, its input and output variables (generalized forces or displacements) allows a very convenient analysis of the phenomenon. We insist on the differences between a phenomenon and a system: the causality, the linearity, the hysteresis are for example properties of phenomena and not properties of system; on the other hand we can consider dissipative or non-dissipative systems. We describe some macroscopic dissipation mechanisms in structures and some microscopic dissipation at the molecular level in materials or mesoscopic dissipation in composites materials. After specifying the notion of internal forces of a system we present some classical dissipative mechanisms currently used: viscous dissipation, friction dissipation, micro-frictions. The purpose of this presentation is not to list new dissipative systems but to point out a number of errors, both scientific and technical, which are frequently committed.

This work, relative to the principle of virtual power, is composed of three distinct but nevertheless complementary parts. The first part follows the line of thought developed by professor Maugin and his students on complex continuous media subject to the objectivity requirement (translational and rotational invariances). The second part shows that this principle is extensible to other types of invariance such as gauge and scale invariances. Gauge invariance allows to express Maxwell equations, usually derived through a vector approach, by use of a scalar principle having the same formal structure as the principle of virtual power. As to scale invariance, it allows to deal, in a general and unified way whatever the underlying physics, with the passage from a continuous medium to a discontinuous one (singular surfaces, lines or points). The third part concerns the foundations of dynamics where the principle of virtual power appears as a theorem, like other analytical principles, each corresponding to one point of view, deductible from a general intrinsic (viewpoint independent) dynamical framework. The attention will be focused on the origin of the duality notion, at the basis of the principle of virtual power.

Dynamic concurrent multiscale modeling methods are reviewed and then analyzed based on their governing equations in terms of consistency in material descriptions between different scales, wave propagation across the numerical interfaces between the different descriptions, and advances in describing defects in the coarse-grained domain. The analysis finds that most methods suffer from the consequences of inconsistent materials descriptions between representations at different scales; a few methods such as Concurrent Atomistic Continuum (CAC), Coupled Atomistic Discrete Dislocation (CADD), and the coupled Extended Finite Element Method (XFEM) are capable of simulating moving defects in the coarse-scale domain to improve practicality and prediction. Application of multiscale simulation to coupled thermal and mechanical problems is showing promise. Mesoscale evolution of defects, largely beyond the reach of conventional atomistic methods, is still beyond the reach of many concurrent multiscale methods.

Growth of semiconductor single crystals under electric and magnetic fields is of interest to increase and better control of crystal growth rate, to suppress and control the adverse effect of natural convection and to obtain better mixing in the growth melt (liquid solution) for better crystal uniformity, which all are favorable conditions for a prolonged growth of high quality crystals. To this end, in parallel to well-designed experiments, modeling is essential to shed light on various aspects of these growth processes and also to better understand the transport phenomena involved. In this article the models developed over the years, mostly based on Professor Gerard Maugin’s well-known contributions to “electromagnetic interactions”, are briefly presented for “solution growth” conducted under electric and magnetic fields. Basic and constitutive equations of a binary electromagnetic continuum mixture are specialized for two important solution growth techniques—Liquid Phase Electroepitaxy (LPEE) and Travelling Heater Method (THM). As an application, an LPEE growth of GaAs bulk crystals under a strong static magnetic field is considered. Experimental results, that have shown that the growth rate under an applied static magnetic field is also proportional to the applied magnetic field and increases with the field intensity level, are predicted from these models. The contribution of a third-order material constant in LPEE is also predicted from these models. The prediction of increasing growth rate in THM growth under rotating magnetic fields from modeling was verified by experiments.

A two-dimensional discrete model for a hexagonal (closed-packed) lattice with elastically interacting round particles possessing two translational and one rotational degrees of freedom is considered. The linear differential-difference equations are obtained by the method of structural modeling to describe propagation of longitudinal, transverse and rotational waves in the medium. The dispersion properties of the model are analyzed. Existence of a backward wave is revealed. The numerical estimations of threshold frequencies of acoustic and rotational waves are given for some values of microstructure parameters.

Dynamics of solitons is considered in the framework of the extended nonlinear Schrödinger equation (NLSE), which is derived from a system of the Zakharov’s type for the interaction between high- and low-frequency (HF and LF) waves, in which the LF field is subject to diffusive damping. The model may apply to the propagation of HF waves in plasmas. The resulting NLSE includes a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term which is well known as an ingredient of the temporal-domain NLSE in optics. Also included is inhomogeneity of the spatial second-order diffraction (SOD). It is shown that the wavenumber downshift of solitons, caused by the pseudo-SRS, may be compensated by an upshift provided by the SOD whose coefficient is a linear function of the coordinate. An analytical solution for solitons is obtained in an approximate form. Analytical and numerical results agree well, including the predicted balance between the pseudo-SRS and the linearly inhomogeneous SOD.

In this paper, we propose an approach for calculating the effective physical properties of composite materials taking into account the percolation phenomena. This approach is based on the Generalized Differential Effective Medium (GDEM) method and, in contrast to the commonly used self-consistent methods, allows us to incorporate the percolation threshold into the homogenization scheme for simulation of the effective elastic moduli and electrical conductivity of a 2D medium. In this case, the composite is treated as a conductive elastic host where elliptical inclusions of two types are embedded: (1) non-conductive soft inclusions and (2) conductive elastic inclusions that have the same properties as the host. The comparison of theoretical simulations with the experimental data for metal plates containing holes has shown that the proposed GDEM approach describes well the elastic moduli and electrical conductivity of materials of such type in the wide range of hole concentration including the area near the percolation threshold.

Among the various types of guided acoustic waves, acoustic wedge waves are non-diffractive and non-dispersive. Both properties make them susceptible to nonlinear effects. Investigations have recently been focused on effects of second-order nonlinearity in connection with anisotropy. The current status of these investigations is reviewed in the context of earlier work on nonlinear properties of two-dimensional guided acoustic waves, in particular surface waves. The role of weak dispersion, leading to solitary waves, is also discussed. For anti-symmetric flexural wedge waves propagating in isotropic media or in anisotropic media with reflection symmetry with respect to the wedge’s mid-plane, an evolution equation is derived that accounts for an effective third-order nonlinearity of acoustic wedge waves. For the kernel functions occurring in the nonlinear terms of this equation, expressions in terms of overlap integrals with Laguerre functions are provided, which allow for their quantitative numerical evaluation. First numerical results for the efficiency of third-harmonic generation of flexural wedge waves are presented.

We analyze the acoustic properties of microstructured repetitive network material undergoing configuration changes leading to geometrical nonlinearities. The effective constitutive law of the homogenized network is evaluated successively as an effective first nonlinear 1D continuum, based on a strain driven incremental scheme written over the reference unit cell, taking into account the changes of the lattice geometry. The dynamical equations of motion are next written, leading to specific dispersion relations. The inviscid Burgers equation is obtained as a specific wave propagation equation for the first order effective continuum when the expression of the energy includes third order contributions, whereas a perturbation method is used to solve the dynamical properties for the effective medium including fourth order terms. This methodology is applied to analyze wave propagation within different microstructures, including the regular and reentrant hexagons, and plain weave textile pattern.

Research on two dimensional (2D) materials, such as Graphene and Molybdenum disulfide (MoS₂), now involves thousands of researchers worldwide, implementing cutting edge technology to study them. Due to the extraordinary properties of 2D materials, research extends from fundamental science to novel applications of 2D materials. This work introduces atomistic simulation methodologies, based on interatomic potential, as a tool to unveil the mechanical and thermal properties at nanoscale of MoS₂, a material that has attracted most research interests among all 2D materials. Young’s modulus, Poison’s ratio, heat conductivity and heat capacity at atomic scale are studied. These findings lend compelling insights into the atomistic mechanism of MoS₂. Then, based on these useful information, we perform concurrent multiscale modeling of MoS₂ from molecular dynamics simulation in atomic region to finite element analysis in continuum region.

A coupled gradient chemoelasticity theory is employed to model the two-phase mechanism that occurs during lithiation of silicon nanoparticles used to fabricate next generation Li-ion battery (LIB) anodes. It is shown that the strain gradient length scale is able to predict the propagation of an interface front of nonzero thickness advancing from the lithiated to unlithiated region without necessarily including higher-order concentration gradients of the Li ions. Larger strain gradient coefficients (elastic internal lengths) induce more diffused interfaces and faster lithiation, which affect both internal strain and stress distributions in a similar way. Estimates for the migration velocity of the phase boundary are obtained and a range of values of the strain gradient length scale is shown to simulate the observed experimental results.

Generalized continuum mechanics (GCM) has attracted increased attention in the context of multiscale materials modeling, an example of which is a bottom-up GCM model, called the atomistic field theory (AFT). Unlike most other GCM models, AFT views a crystalline material as a continuous collection of lattice points; embedded within each point is a unit cell with a group of discrete atoms. As such, AFT concurrently bridges the discrete and continuous descriptions of materials, two fundamentally different viewpoints. In this chapter, we first review the basics of AFT and illustrate how it is realized through coarse-graining atomistic simulations via a concurrent atomistic-continuum (CAC) method. Important aspects of CAC, including its advantages relative to other multiscale methods, code development, and numerical implementations, are discussed. Then, we present recent applications of CAC to a number of metal plasticity problems, including static dislocation properties, fast moving dislocations and phonons, as well as dislocation/grain boundary interactions. We show that, adequately replicating essential aspects of dislocation fields at a fraction of the computational cost of full atomistics, CAC is established as an effective tool for coarse-grained modeling of various nano/micro-scale thermal and mechanical problems in a wide range of monatomic and polyatomic crystalline materials.

This paper presents a theoretical analysis on the bending and shear of a cantilever ZnO piezoelectric semiconductor fiber under a transverse end force. The phenomenological theory of piezoelectric semiconductors consisting of Newton’s second law of motion, the charge equation of electrostatics, and the conservation of charge of electrons and holes is used. The equations are linearized for a small end force and small electromechanical fields as well as small carrier concentration perturbations. A first-order, one-dimensional theory for the bending of ZnO fibers with shear deformation is derived from the linearized three-dimensional equations. An analytical solution is obtained. The electromechanical fields and carrier concentrations are calculated. It is found that the electric potential is nearly constant along the fiber except near the fixed end of the cantilever, and that the electron distribution over a cross section is due to the transverse shear force and the piezoelectric constant e₂₄.

The purpose of this work is to present general solutions for two-dimensional (2D) plane-strain contact problems within the framework of the generalized continuum theory of couple-stress elasticity. This theory is able to capture the scale effects, which are often observed in indentation problems with contact lengths comparable to the material microstructure. To this end, we formulate a number of basic contact problems in terms of singular integral equations using the pertinent Green’s function that corresponds to the solution of the analogue of the Flamant-Boussinesq problem of a half-space in couple-stress elasticity. In addition, we also provide results concerning the more complex traction boundary-value problem involving a deformable layer (again within couple-stress elasticity) of finite thickness superposed on a rigid half-space. We show that the contact behavior of materials with couple-stress effects depends strongly upon their microstructural characteristics, especially when the characteristic dimension of the microstructure becomes comparable to macroscopic characteristic dimensions of the contact problem. The latter lengths could be either the contact length/area or even the thickness of the layer.

The scattering of incident surface waves by a cylindrical cavity of arbitrary shape near the free surface of an elastic half-space is considered in this paper. The scattered field is represented by the radiation from equivalent body forces. The equivalent body forces due to the horizontal and vertical displacement components of the incident surface wave are determined separately. It is found that the equivalent body forces are double forces parallel and normal to the free surface of the half-space. By the use of the elastodynamic reciprocity theorem, the surface waves generated by the equivalent double forces are obtained in terms of properties of the incident wave, the cross-sectional area of the cavity and the elastic constants of the elastic half-space. The superposition of the surface waves generated by the equivalent body forces represents the scattered field of surface waves.