Infinitesimal Volume Element Research Articles

Stress average rule derived through the principle of virtual power

AbstractStress and strain average rules are the key conceptual pillars of the wide field of continuum micromechanics of materials. The aforementioned rules express that the spatial average of (micro‐)stress and (micro‐)strain fields throughout a microscopically finite representative volume element (RVE) are equal to the (macro‐)stress and (macro‐)strain values associated with the corresponding macroscopically infinitesimal volume element (macroscopic material point). According to the famous contribution of Hashin, stress and strain average rules are derived from equilibrium and compatibility conditions, together with (micro‐)displacement and (micro‐)traction boundary conditions associated with homogeneous (macro‐)strains and (macro‐)stresses, respectively. However, as, strictly speaking, only displacements or tractions can be described at the boundary, the remaining average rule turns out as a mere definition. We here suggest a way to do without such a definition, by resorting to the principle of virtual power (PVP) as a means to guarantee mechanical equilibrium: at the boundary of the RVE, we prescribe virtual (micro‐)velocities, which are linked to arbitrary, but homogeneous virtual (macro‐)velocities and (macro‐)strain rates, while the latter are also linked, in a multilinear fashion, with the microscopic virtual strain rate fields inside the RVE. Considering, under these conditions, equivalence of the macroscopic and the microscopic expressions for the virtual power densities of the internal and the external forces yields the well‐known stress average rule and, in case of microscopically uniform force fields, a volume force average rule. The same strategy applied to an RVE hosting single forces between atomistic mass points, readily yields the macroscopic “internal virial stress tensor.”

Electron Density Geometry and the Quantum Theory of Atoms in Molecules.

A novel form of charge density analysis, that of isosurface curvature redistribution, is formulated and applied to the toy problem of carbonyl oxygen activation in formaldehyde. The isosurface representation of the electron charge density allows us to incorporate the rigorous geometric constraints of closed surfaces toward the analysis and chemical interpretation of the charge density response to perturbations. Visual inspection of 2D isosurface motion resulting from applied external electric fields reveals how the isosurface curvature flows within and between atoms and that a molecule can be uniquely and completely partitioned into chemically significant regions of positive and negative curvatures. These concepts reveal that carbonyl oxygen activation proceeds primarily through curvature and charge redistribution within rather than between Bader atoms. Using gradient bundle analysis─the partitioning of formaldehyde into infinitesimal volume elements bounded by QTAIM zero-flux surfaces─the observations from visual isosurface inspection are verified. The results of the formaldehyde carbonyl analysis are then shown to be transferable to the substrate carbonyl in the ketosteroid isomerase enzyme, laying the groundwork for extending this approach to the problems of enzymatic catalysis.

Generalization and Modelling of Rigid Polyisocyanurate Foam Reaction Kinetics, Structural Units Effect, and Cell Configuration Mechanism

PIR/PUR ratio was derived from differential manipulation of generalized polyisocyanurate kinetic model. The structural unit effects on polymerization of isocyanurate, urethane and urea linkages were evaluated based on Mayo-Lewise tercopolymerization scheme. The cell microstructural configuration model was further developed from profiled FOAMAT reactivity parameters with integrated analysis of cell interface physics. The interstitial border area was defined by interface free energy theory, the shear viscosity was evaluated by foam motion, gas fraction, and partial pressure, and the cell inflation was re-examined by gas-liquid surface tension variability. The cell anisotropic degree, assumed as an aspect ratio of infinitesimal volume elements in cell uniformity, was characterized by equilibrated work increase of surface energy approximated by 2D stretching deformation from sphere cell to spheroid cell. The relationship between pressure and surface tension of elongated cells was also derived from modelling at the same condition of cell deformation.

An alternative perspective on the concept of stress in classical continuum mechanics

AbstractWithin a variational formulation of continuum mechanics, as proposed for instance by Germain [1], the internal virtual work contribution of a continuum is postulated as a smooth density integrated over the deformed configuration of the body. In this smooth density the stress field appears as dual quantity to the gradient of the virtual displacement field. Since the mathematical definition of the volume integral naturally provides a macro‐micro relation between infinitesimal volume elements and the continuous body, we propose in this paper an alternative definition of stress on the micro level of the infinitesimal volume elements. In particular, the stress is defined as the internal forces of the body that model the mutual force interaction between neighboring volume elements. The existence of the stress tensor on the macro level is then obtained from the summation of all virtual work contributions within the body, followed by a limit process in which the volume elements are sent to zero. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Quantum continuum mechanics in a strong magnetic field

We extend a recent formulation of quantum continuum mechanics [J. Tao et. al, Phys. Rev. Lett. {\bf 103}, 086401 (2009)] to many-body systems subjected to a magnetic field. To accomplish this, we propose a modified Lagrangian approach, in which motion of infinitesimal volume elements of the system is referred to the "quantum convective motion" that the magnetic field produces already in the ground-state of the system. In the linear approximation, this approach results in a redefinition of the elastic displacement field $\uv$, such that the particle current $\jv$ contains both an electric displacement and a magnetization contribution: $\jv=\jv_0+n_0\partial_t \uv+\nabla \times (\jv_0\times \uv)$, where $n_0$ and $\jv_0$ are the particle density and the current density of the ground-state and $\partial_t$ is the partial derivative with respect to time. In terms of this displacement, we formulate an "elastic approximation" analogous to the one proposed in the absence of magnetic field. The resulting equation of motion for $\uv$ is expressed in terms of ground-state properties -- the one-particle density matrix and the two-particle pair correlation function -- and in this form it neatly generalizes the equation obtained for vanishing magnetic field.

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.The semi-classical analysis presented in this paper does away with most of the ambiguities found in the initial semi-finite spectral triple construction. The cubic lattices play the role of a coordinate system and a divergent sequence of free parameters found in the Dirac type operator is identified as a certain inverse infinitesimal volume element.

Elastic waves propagation in 1D fractional non-local continuum

Aim of this paper is the study of waves propagation in a fractional, non-local 1D elastic continuum. The non-local effects are modeled introducing long-range central body interactions applied to the centroids of the infinitesimal volume elements of the continuum. These non-local interactions are proportional to a proper attenuation function and to the relative displacements between non-adjacent elements. It is shown that, assuming a power-law attenuation function, the governing equation of the elastic waves in the unbounded domain, is ruled by a Marchaud-type fractional differential equation. Wave propagation in bounded domain instead involves only the integral part of the Marchaud fractional derivative. The dispersion of elastic waves, as well as waves propagation in unbounded and bounded domains are discussed in detail.

Gravity field of Jupiter’s moon Amalthea and the implication on a spacecraft trajectory

Before its final plunge into Jupiter in September 2003, GALILEO made a last ‘visit’ to one of Jupiter’s moons – Amalthea. This final flyby of the spacecraft’s successful mission occurred on November 5, 2002. In order to analyse the spacecraft data with respect to Amalthea’s gravity field, interior models of the moon had to be provided. The method used for this approach is based on the numerical integration of infinitesimal volume elements of a three-axial ellipsoid in elliptic coordinates. To derive the gravity field coefficients of the body, the second method of Neumann was applied. Based on the spacecraft trajectory data provided by the Jet Propulsion Laboratory, GALILEO’s velocity perturbations at closest approach could be calculated. The harmonic coefficients of Amalthea’s gravity field have been derived up to degree and order six, for both homogeneous and reasonable heterogeneous cases. Founded on these numbers the impact on the trajectory of GALILEO was calculated and compared to existing Doppler data. Furthermore, predictions for future spacecraft flybys were derived. No two-way Doppler-data was available during the flyby and the harmonic coefficients of the gravity field are buried in the one-way Doppler-noise. Nevertheless, the generated gravity field models reflect the most likely interior structure of the moon and can be a basis for further exploration of the Jovian system.

On the Possibility of Elastic Strain Localisation in a Fault

The phenomenon of strain localisation is often observed in shear deformation of particulate materials, e.g., fault gouge. This phenomenon is usually attributed to special types of plastic behaviour of the material (e.g., strain softening or mismatch between dilatancy and pressure sensitivity or both). Observations of strain localisation in situ or in experiments are usually based on displacement measurements and subsequent computation of the displacement gradient. While in conventional continua the symmetric part of the displacement gradient is equal to the strain, it is no longer the case in the more realistic descriptions within the framework of generalised continua. In such models the rotations of the gouge particles are considered as independent degrees of freedom the values of which usually differ from the rotation of an infinitesimal volume element of the continuum, the latter being described for infinitesimal deformations by the non-symmetric part of the displacement gradient. As a model for gouge material we propose a continuum description for an assembly of spherical particles of equal radius in which the particle rotation is treated as an independent degree of freedom. Based on this model we consider simple shear deformations of the fault gouge. We show that there exist values of the model parameters for which the displacement gradient exhibits a pronounced localisation at the mid-layers of the fault, even in the absence of inelasticity. Inelastic effects are neglected in order to highlight the role of the independent rotations and the associated additional parameters. The localisation-like behaviour occurs if (a) the particle rotations on the boundary of the shear layer are constrained (this type of boundary condition does not exist in a standard continuum) and (b) the contact moment-or bending stiffness is much smaller than the product of the effective shear modulus of the granulate and the square of the width of the gouge layer. It should be noted however that the virtual work functional is positive definite over the range of physically meaningful parameters (here: contact stiffnesses, solid volume fraction and coordination number) so that strictly speaking we are not dealing with a material instability.

Non-linear static–dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain–displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.

A variational approach to charged spinning fluids in general relativity

In this paper we present a variational model for a charged spinning fluid in interaction with the gravitational field. This model is a natural extension of a previous one dealing with neutral spinning fluids already studied by the authors, in which each infinitesimal volume element is considered as being a replica of a micro-rigid body endowed with rotational inertia.

Spinning fluids in the Einstein-Cartan theory: A variational formulation

In this work we propose a lagrangian for spinning fluids in the Einstein-Cartan theory. The basic characteristic of the model is to consider each infinitesimal volume element as replicas of micro-rigid bodies. The theory obtained represents the thermodynamical equilibrium limit of a more general situation where dissipative processes due to spin take place. We outline the extension of such processes to the Einstein-Cartan theory.

A dynamic model for a Timoshenko beam in an elastic-plastic state

In this paper, equations which describe the propagation of elastic-plastic waves of combined stress resultants in a Timoshenko beam under symmetrical bending and tension or compression are derived. It is assumed that the beam material exhibits a weak work-hardening and strain-rate independent behaviour. The normality rule is postulate for the plastic strain rates of an infinitesimal volume element. The wave propagation is described by a hyperbolic system of differential equations, the state variables of which are the stress resultants and the strain rates of an infinitesimal beam element. It is shown that the actual distribution of shear stress in the plastic range need not be taken into account by means of a plastic shear correction factor.

A kinetic model for ideal plug-flow reactors

Simultaneous differential equations of plug-flow reactors resulting from mass balances on substrate and biomass around an infinitesimal volume element are solved analytically taking the longitudinal biomass gradient into account under steady-state conditions. A relationship between substrate and biomass concentrations and an analytical solution for substrate and/or biomass concentration as a function of hydraulic residence time are developed. Design examples are given and these have shown that results of analytical solution are in good agreement with those of differential equations obtained by the finite difference method. Results of this work may help engineers to acquire a new understanding about the plug-flow reactors.

Information theory lateral density distribution for Earth inferred from global gravity field

Information theory inference, better known as the maximum entropy method, is used to infer the lateral density distribution inside the earth. The approach assumes that the earth consists of indistinguishable Maxwell‐Boltzmann particles populating infinitesimal volume elements and follows the standard methods of statistical mechanics (maximizing the entropy function). The GEM 10B spherical harmonic gravity field coefficients, complete to degree and order 36, are used as constraints on the lateral density distribution. The spherically symmetric part of the density distribution is assumed to be known. The lateral density variation is assumed to be small in comparison with the spherically symmetric part. The resulting information theory density distribution for the cases of no crust removed, 30 km of compensated crust removed, and 30 km of uncompensated crust removed all give broad density anomalies extending deep into the mantle, but with the density contrasts being the greatest toward the surface (typically ±0.004 g cm−3 in the first two cases and ±0.04 g cm−3 in the third). None of the density distributions resemble classical organized convection cells. The information theory approach may have use in choosing standard earth models, but the inclusion of seismic data into the approach appears difficult.

Oxygen diffusion in large single-celled organisms.

The functional relationship between the oxygen uptake rate of a spherical, single cell organism and the external oxygen tension is shown to be related to the dependence of the specific oxygen consumption rate, that is, the consumption rate of an infinitesimal volume element of cellular material, on the external oxygen tension. Analytical solutions of the governing steady state diffusion equation are obtained by dividing the system into three regions, an inner region of the sphere in which oxygen consumption rate depends upon oxygen tension, an outer region of the sphere in which oxygen consumption rate is constant (independent of oxygen tension), a nonconsuming membrane over the sphere that offers only resistance to oxygen diffusion, and an infinite region outside the sphere and membrane supplying oxygen to the system. The solutions show the oxygen tension as a function of position inside the spherical cell for a variety of system parameters.