A material frame approach for evaluating continuum variables in atomistic simulations

We present a material frame formulation analogous to the spatial frame formulation developed by Hardy, whereby expressions for continuum mechanical variables such as stress and heat flux are derived from atomic-scale quantities intrinsic to molecular simulation. This formulation is ideally suited for developing an atomistic-to-continuum correspondence for solid mechanics problems. We derive expressions for the first Piola–Kirchhoff (P–K) stress tensor and the material frame heat flux vector directly from the momentum and energy balances using localization functions in a reference configuration. The resulting P–K stress tensor, unlike the Cauchy expression, has no explicit kinetic contribution. The referential heat flux vector likewise lacks the kinetic contribution appearing in its spatial frame counterpart. Using a proof for a special case and molecular dynamics simulations, we show that our P–K stress expression nonetheless represents a full measure of stress that is consistent with both the system virial and the Cauchy stress expression developed by Hardy. We also present an expanded formulation to define continuum variables from micromorphic continuum theory, which is suitable for the analysis of materials represented by directional bonding at the atomic scale.     

Mechano-chemistry; diffusion in multicomponent compressible mixtures

In the present work we derive the volume continuity equation and demonstrate its use to define the volume frame of reference in the multicomponent, compressible systems. The volume velocity (material velocity) is a unique frame of reference for all internal forces and processes, e.g., the mass diffusion, heat flow, etc. No basic changes are required in the foundations of linear irreversible thermodynamics except recognizing the need to add volume to the usual list of extensive physical properties undergoing transport in every continuum. The volume fixed frame of reference allows the translation of the Newton’s discrete mass-point molecular mechanics into continuum mechanics and the use of the Cauchy linear momentum equation of fluid mechanics and Navier-Lamé equation of mechanics of solids.Our proposed modifications of Navier-Lamé and energy conservation equations are self-consistent with the literature for solid-phase continua dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken in terms of diffuse volume transport. We do show that the local diffusion processes do not change the centre of mass of the system and that the stress and viscosity depend only on the local volume velocity.     

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.     

Some applications of geometry is continuum mechanics

Some contemporary ideas from differential geometry are applied to continuum mechanics. The Lie derivative is used to clarify the notion of “objective rates”, an intrinsic treatment of Piola transformations is described, a simplified proof of Vainberg's theorem for potential operators is given by way of the Poincaré lemma on infinite dimensional manifolds, and a new derivation of the basic equations of continuum mechanics is presented which is valid in a general Riemannian manifold setting.     

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.     

Length scales at which classical elasticity breaks down for various materials

At what characteristic length scale does classical continuum elasticity cease to accurately describe small deformation mechanical behavior? The two dominant physical mechanisms that lead to size dependency of elastic behavior at the nanoscale are surface energy effects and nonlocal interactions. The latter arises due to the discrete structure of matter and the fluctuations in the interatomic forces that are smeared out within the phenomenological elastic modulus at coarser sizes. While surface energy effects have been well characterized in the literature, little is known about the length scales at which nonlocal effects manifest for different materials. Using a combination of empirical molecular dynamics and lattice dynamics (empirical and ab initio), we provide estimates of nonlocal elasticity length scales for various classes of materials: semiconductors, metals, amorphous solids, and polymers.     

Analytical expression of transition form factors from discrete spectrum nS states to continuum states of hydrogen-like atoms

The new approach to calculation of transition form factors of hydrogen-like atoms is proposed. The explicit analytical expressions of transition form factors from arbitrary discrete states to continuum are obtained in terms of classical polynomials.     

A discrete negative AKNS equation: generalized Cauchy matrix approach

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit we obtain a semidiscrete negative AKNS equation and after a second continuum limit we derive the nonlinear negative AKNS equation. The reductions to discrete, semi-discrete and continuous sine-Gordon equations are also discussed.     

Finite element analysis of the demagnetization effect and stress inhomogeneities in magnetic shape memory alloy samples

This paper is concerned with the finite element analysis of boundary value problems involving nonlinear magnetic shape memory behavior, as might be encountered in experimental testing or engineering applications of magnetic shape memory alloys (MSMAs). These investigations mainly focus on two aspects: first, nonlinear magnetostatic analysis, in which the nonlinear magnetic properties of the MSMA are predicted by the phenomenological internal variable model previously developed by Kiefer and Lagoudas, is utilized to investigate the influence of the demagnetization effect on the interpretation of experimental measurements. An iterative procedure is proposed to deduce the true constitutive behavior of MSMAs from experimental data that typically reflect the shape-dependent system response of a sample. Secondly, the common assumption of a homogeneous Cauchy stress distribution in the MSMA sample is tested. This is motivated by the expectation that the influence of magnetic body forces and body couples caused by field matter interactions may not be negligible in MSMAs that exhibit blocking stresses of well below 10?MPa. To this end, inhomogeneous Maxwell stress distributions are first computed in a post-processing step, based on the magnetic field and magnetization distributions obtained in the magnetostatic analysis. Since the computed Maxwell stress fields, though allowing a first estimation of the influence of the magnetic force and couple, do not satisfy equilibrium conditions, a finite element analysis of the coupled field equations is performed in a second step to complete the study. It is found that highly non-uniform Cauchy stress distributions result under the influence of magnetic body forces and couples, with magnitudes of the stress components comparable to externally applied bias stress levels.     

Exact solutions of a new, 2D frictionless contact model for orthotropic piezoelectric materials indented by a rigid sliding punch

A theoretical analysis of two-dimensional frictionless sliding contact over orthotropic piezoelectric materials indented by a rigid sliding punch is carried out using a real fundamental solution approach. The actual sliding motion does occur, which is different from the classical sliding contact, and the Galilean transformation is introduced to make the governing equations containing the inertial terms tractable. A system of Cauchy singular integral equations is derived and exact solutions are obtained for the cases of a conducting flat punch and a cylindrical punch, respectively. Explicit expressions of various stresses and electric displacement for each case of eigenvalue distribution of the corresponding characteristic equation are obtained. Numerical results are presented to justify the validity of exact solutions. The effects of various mechanical-electric and geometrical loadings, dimensionless sliding speed and punch foundation profiles on the surface contact stress, surface electric charge and surface in-plane stress are presented. The singular behaviors at the edges of the punch are also revealed.     

On the flexoelectric deformations of finite size bodies

Exact equations describing flexoelectric deformation in solids, derived previously within the framework of a continuum media theory, are partial differential equations of the fourth order. They are too complex to be used in the cases interesting for applications. In this paper, using the fact of smallness of the elastic moduli of a higher order, simplified equations are proposed. Solution of the exact equations is approximately represented as a sum of two parts: the first part obeys one-dimensional differential equations and exponentially decays near surface and the second part satisfies the equations of classical theory of elasticity. The first part can be constructed in an explicit form. For the second part, boundary conditions are obtained. They have a form of the classical boundary conditions for the body under external forces on surface.     

分层结构中兰姆波二次谐波发生的模式展开分析

采用二阶微扰近似和模式展开方法,对分层结构中兰姆波的二次谐波发生问题进行了深入研究,得到了二次谐波声场的一般表达式。兰姆波在分层结构中传播时,固体板材的体弹性非线性效应导致在各固体层的内部及表面出现二倍频的体驱动力及面驱动应力,其作用是在分层结构中激发出一系列二倍频兰姆波的简正模式,这些简正模式的叠加即构成分层结构中兰姆波的二次谐波声场。当构成二次谐波声场的某一二倍频兰姆波简正模式的相速度等于基频兰姆波的相速度时,该二倍频兰姆波简正模式分量表现出随传播距离积累增长的性质。本文还就铝板-环氧粘接层-铝板结构中兰姆波的二次谐波声场进行了数值分析,结果显示,软粘接层性质对二次谐波声场的影响很大。     

Elastic constitutive laws for incompatible crystalline media: the contributions of dislocations,disclinations and G-disclinations

Linear higher-grade higher-order elastic constitutive laws for compatible (defect-free) and incompatible (containing crystal line defects) media are presented. In the proposed model, the free energy density of a body subjected to elastic deformation under the action of surface tractions, moments or hyper-traction tensors (second-order tensors whose anti-symmetric part corresponds to moments) has contributions coming from the first two gradients of displacements. Thermodynamic considerations reveal that only the symmetric component of the gradient of elastic displacement, i.e., compatible elastic strain tensor, and the anti-symmetric component of the second gradient of elastic displacement, i.e., compatible third-order elastic curvature tensor, contribute to the free energy density during compatible deformation of the body. The line crystal defect contributions are accounted for by incorporating the incompatible components of elastic strains, curvatures and symmetric 2-distortions as state variables of the free energy density. In particular, the presence of generalized disclinations (G-disclinations) is acknowledged when the medium is subjected to surface hyper-traction tensors having a non-zero symmetric component along with surface-tractions on its boundary. Mechanical dissipation analysis provides for the coupling between the Cauchy stresses and third-order symmetric hyper-stresses. The free energy density and elastic laws for a defect-free and line crystal defected medium are proposed in a linear setting. In the special case of isotropy, the cross terms between elastic strains and curvatures contribute to the free energy density through a single elastic constant. More interestingly, the Cauchy and couple stresses are found to have contributions coming from both, elastic strains and curvatures.     

A fractional-order Darcy's law

By using spatial averaging methods, in this work we derive a Darcy's-type law from a fractional Newton's law of viscosity, which is intended to describe shear stress phenomena in non-homogeneous porous media. As a prerequisite towards this end, we derive an extension of the spatial averaging theorem for fractional-order gradients. The usage of this tool for averaging continuity and momentum equations yields a Darcy's law with three contributions: (i) similar to the classical Darcy's law, a term depending on macroscopic pressure gradients and gravitational forces; (ii) a fractional convective term induced by spatial porosity gradients; and (iii) a fractional Brinkman-type correction. In the three cases, the corresponding permeability tensors should be computed from a fractional boundary-value problem within a representative cell. Consistency of the resulting Darcy's-type law is demonstrated by showing that it is reduced to the classical one in the case of integer-order velocity gradients and homogeneous porous media.     

Mode-I crack in a two-dimensional fibre-reinforced generalized thermoelastic problem

A general model of the equations of the Lord-Şulman theory including one relaxation time and the Green-Lindsay theory with two relaxation times, as well as the classical dynamical coupled theory, are applied to the study of the influence of reinforcement on the total deformation for an infinite space weakened by a finite linear opening mode-I crack. We study the influence of reinforcement on the total deformation of rotating thermoelastic half-space and their interaction with each other. The material is homogeneous isotropic elastic half space. The crack is subjected to prescribed temperature and stress distributions. The normal mode analysis is used to obtain the exact expressions for displacement components, force stresses, and temperature. The variations of the considered variables with the horizontal distance are illustrated graphically. Comparisons are made with the results obtained in the three theories with and without rotation. A comparison is also made between the two theories for different depths.     

The effect of rotation on plane waves in generalized thermo-microstretch elastic solid with one relaxation time for a mode-I crack problem

The present paper is aimed at studying the effect of rotation on the general model of the equations of the generalized thermo-microstretch for a homogeneous isotropic elastic half-space solid,whose surface is subjected to a Mode-I crack problem.The problem is studied in the context of the generalized thermoelasticity Lord-S hulman’s (L-S) theory with one relaxation time,as well as with the classical dynamical coupled theory (CD).The normal mode analysis is used to obtain the exact expressions for the displacement components,the force stresses,the temperature,the couple stresses and the microstress distribution.The variations of the considered variables through the horizontal distance are illustrated graphically.Comparisons of the results are made between the two theories with and without the rotation and the microstretch constants.     

Analysis of axisymmetric shell structures under axisymmetric loading by the flexibility method

A method is developed for the static stress and deformation analysis of axisymmetric shells under axisymmetric loading by reduction of the shell to ring sections. In particular, the wall thickness of the shell may vary and the method is applicable to the analysis of shells with irregular meridional geometry. Explicit expressions for the influence coefficients for each ring element are derived. In the development of these expressions, exact evaluation of stresses in the circumferential direction of the ring is used. The distribution of stresses in the meridional direction of the ring element is assumed to be linear with each element. By using the derived influence coefficients, the unknown forces at the junctures of the ring elements are found by the standard flexibility method of indeterminate structural analysis. Subsequently, the displacements and internal stresses are determined. Example solutions for a flat circular plate under transverse loading and for a cylindrical shell under a boundary edge loading show excellent agreement with solutions found by solving the governing differential equations.