Continuum Formulation Research Articles

Structural biology response of a collagen hydrogel synthetic extracellular matrix with embedded human fibroblast: computational and experimental analysis

Adherent cells exert contractile forces which play an important role in the spatial organization of the extracellular matrix (ECM). Due to these forces, the substrate experiments a volume reduction leading to a characteristic shape. ECM contraction is a key process in many biological processes such as embryogenesis, morphogenesis and wound healing. However, little is known about the specific parameters that control this process. With this aim, we present a 3D computational model able to predict the contraction process of a hydrogel matrix due to cell-substrate mechanical interaction. It considers cell-generated forces, substrate deformation, ECM density, cellular migration and proliferation. The model also predicts the cellular spatial distribution and concentration needed to reproduce the contraction process and confirms the minimum value of cellular concentration necessary to initiate the process observed experimentally. The obtained continuum formulation has been implemented in a finite element framework. In parallel, in vitro experiments have been performed to obtain the main model parameters and to validate it. The results demonstrate that cellular forces, migration and proliferation are acting simultaneously to display the ECM contraction.

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.

On bounds and boundary conditions in the continuum Landau gauge

In this note, we consider the Landau gauge in the continuum formulation. Our purposes are twofold. Firstly, we try to work out the consequences of the recently derived Cucchieri–Mendes bounds on the inverse Faddeev operator at the level of the path integral quantization. Secondly, we give an explicit (renormalizable) prescription to implement the so-called Landau \(B\)-gauges as introduced by Maas.

Asymptotic analysis of a noncontact AFM microcantilever sensor with external feedback control

An external feedback control is inserted in a nonlinear continuum formulation of a noncontact AFM model. The aim of the feedback is to keep the system response to an operationally suitable one, thus allowing reliable measurement of the sample surface by avoiding possible unstable microcantilever sensor motions. The study of the weakly nonlinear system dynamics about the desired fixed point close to primary resonance is carried out via multiple-scale asymptotics, whose outcomes are validated via numerical simulations of the original system equations of motion. The latter include controllable periodic dynamics and additional periodic and distinct quasiperiodic solutions that appear beyond the asymptotic stability thresholds. The results highlight the effectiveness of the applied feedback control technique and also enable the derivation of a comprehensive system bifurcation structure highlighting the stability thresholds for robust controllable AFM dynamics.

Towards Phase Field Modeling of Ductile Fracture in Gradient‐Extended Elastic‐Plastic Solids

AbstractThe modeling of failure in ductile metals must account for complex phenomena at a micro‐scale as well as the final rupture at the macro‐scale. Within a top‐down viewpoint, this can be achieved by the combination of a micro‐structure‐informed elastic‐plastic model with a concept for the modeling of macroscopic crack discontinuities. In this context, it is important to account for material length scales and thermo‐mechanical coupling effects due to dissipative heating. This can be achieved by the construction of non‐standard, gradient‐enhanced models of plasticity with a full embedding into continuum thermodynamics [1,2]. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture based on regularized crack discontinuities. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns within a pure continuum formulation. Moreover, the phase field modeling of fracture is related to gradient theories of continuum damage mechanics, and fits nicely the structure of constitutive models for gradient plasticity. The main focus of this work is the extensions to gradient thermoplasticity and phase field formulation of ductile fracture, conceptually in line with the work [3]. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A consistent u‐p formulation for porous media with hysteresis

SummaryThis paper presents a continuum formulation based on the theory of porous media for the mechanics of liquid unsaturated porous media. The hysteresis of the liquid retention model is carefully modelled, including the derivation of the corresponding consistent tangent moduli. The quadratic convergence of Newton's method for solving the highly nonlinear system with an implicit finite element code is demonstrated. A u‐p formulation is proposed where the time discretisation is carried out prior to the space discretisation. In this way, the derivation of all consistent moduli is fairly straightforward. Time integration is approximated with the Theta and Newmark's methods, and hence the fully coupled nonlinear dynamics of porous media is considered. It is shown that the liquid retention model requires also the consistent second‐order derivative for quadratic convergence. Some predictive simulations are presented illustrating the capabilities of the formulation, in particular to the modelling of complex porous media behaviour. Copyright © 2014 John Wiley & Sons, Ltd.

Electronic Excitations in Nonpolar Solvents: Can the Polarizable Continuum Model Accurately Reproduce Solvent Effects?

In nonpolar solvents, both electrostatic and nonelectrostatic interactions play a role in tuning the electronic excitations of molecular solutes. This specificity makes the application of continuum solvation models a challenge. Here, we propose a strategy for the calculation of solvatochromic shifts on absorption spectra, using a coupling of the polarizable continuum model with a time-dependent density functional theory framework, which explicitly accounts for dispersion and repulsion, as well as for electrostatic effects. Our analysis makes a step further in the interpretation of the effects of nonpolar solvents and suggests new directions in their modeling using continuum formulations.

Atomic diffusion theory challenging the Cahn-Hilliard method

The author considers a theory of diffusion in nonuniform alloys at the atomic level going beyond the uncorrelated Cahn-Hilliard approximation. A surprising conclusion is reached that a thermodynamic (i.e. continuum) formulation of diffusion is impossible.

Dipole formation and yielding in a two-dimensional continuum dislocation model

While a Taylor-type yield stress, proportional to the square-root of the dislocation density, may appear at a macroscopic scale, it can be shown with discrete dislocation simulations that it does not accurately describe dislocation motion on the scale of individual dislocations. In this article, we first demonstrate that the Taylor term fails to capture a number of features of dislocation dynamics by comparing the results of a continuum formulation using this yield stress term to discrete dislocation dynamics simulations. We then present an alternate model, based on a mean free path formulation, and demonstrate that this model effectively reproduces the results of the discrete simulation. This mean free path model is proposed as an extension to an existing continuum theory, making use of the key fact that the velocity of dislocations in a realistic system is not single valued, but distributed over several values. The velocity distribution may also change with time. It is demonstrated that this formulation predicts features of both pileups and homogenous distributions which are in agreement with discrete dislocation simulations, but are not reproducible by traditional statistical continuum theories. Finally, possible extensions to this model are discussed, which may enhance the ability to reproduce key features of dislocation yielding.

From coherent to incoherent mismatched interfaces: A generalized continuum formulation of surface stresses

The equilibrium of coherent and incoherent mismatched interfaces is reformulated in the context of continuum mechanics based on the Gibbs dividing surface concept. Two surface stresses are introduced: a coherent surface stress and an incoherent surface stress, as well as a transverse excess strain. The coherent surface stress and the transverse excess strain represent the thermodynamic driving forces of stretching the interface while the incoherent surface stress represents the driving force of stretching one crystal while holding the other fixed and thereby altering the structure of the interface. These three quantities fully characterize the elastic behavior of coherent and incoherent interfaces as a function of the in-plane strain, the transverse stress and the mismatch strain. The isotropic case is developed in detail and particular attention is paid to the case of interfacial thermo-elasticity. This exercise provides an insight on the physical significance of the interfacial elastic constants introduced in the formulation and illustrates the obvious coupling between the interface structure and its associated thermodynamics quantities. Finally, an example based on atomistic simulations of Cu/Cu2O interfaces is given to demonstrate the relevance of the generalized interfacial formulation and to emphasize the dependence of the interfacial thermodynamic quantities on the incoherency strain with an actual material system.

One-dimensional action for simplicial gravity in three dimensions

This article presents a derivation of the Ponzano--Regge model from a one-dimensional spinor action. The construction starts from the first-order Palatini formalism in three dimensions. We then introduce a simplicial decomposition of the three-dimensional manifold and study the discretised action in the spinorial representation of loop gravity. A one-dimensional refinement limit along the edges of the discretisation brings us back to a continuum formulation. The three-dimensional action turns into a line integral over the one-skeleton of the simplicial manifold. All fields are continuous but have support only along the one-dimensional edges. We define the path integral, and remove the redundant integrals over the local gauge orbits through the usual Faddeev--Popov procedure. The resulting state sum model reproduces the Ponzano--Regge amplitudes.

Polarization in dielectrics modeled as micromorphic continua

On the basis of an extension of the micromorphic continuum theory accounting for electromagneto-elastic coupling in a consistent way, we develop a continuum formulation of polarization in dielectrics. The introduction of dielectric multipoles by means of the microdisplacement and the exploitation of their linearized evolution equations allows to achieve a representation of the polarization in terms of macroscopic and microscopic strain measures and in particular of the microrotation tensor. The microdeformation is then expressed by means of the macroscopic strain and the dependence of polarization from the gradients of the mechanical displacement is explicitly written, up to the third order in multipoles. Piezoelectric and flexoelectric tensors are thus obtained, and it is shown that they uniquely depend on electric quadrupoles and octupoles. We also discuss the reduction in the present model to micropolar continua, giving a representation of the flexoelectric tensor for isotropic micropolar dielectrics.

Covariant gauges without Gribov ambiguities in Yang-Mills theories

We propose a one-parameter family of nonlinear covariant gauges which can be formulated as an extremization procedure that may be amenable to lattice implementation. At high energies, where the Gribov ambiguities can be ignored, this reduces to the Curci-Ferrari-Delbourgo-Jarvis gauges. We further propose a continuum formulation in terms of a local action which is free of Gribov ambiguities and avoids the Neuberger zero problem of the standard Faddeev-Popov construction. This involves an averaging over Gribov copies with a nonuniform weight, which introduces a new gauge-fixing parameter. We show that the proposed gauge-fixed action is perturbatively renormalizable in four dimensions and we provide explicit expressions of the renormalization factors at one loop. We discuss the possible implications of the present proposal for the calculation of Yang-Mills correlators.

Geometric model of a fullerene molecule in the presence of Aharonov–Bohm flux

In this study, we describe a geometric model of a fullerene molecule with Ih symmetry. We combine the well known non-Abelian monopole approach and the geometric theory of defects, where every topological defect is associated with curvature and torsion, to describe a fullerene molecule. The geometric theory of defects in solids is used to consider the topological defects that allow this molecule to form and we apply a continuum formulation to describe this spherical geometry in the presence of an external Aharonov–Bohm flux. We solve a Dirac equation for this model and obtain the eigenvalues and eigenfunction of the Hamiltonian, and we obtain the persistent current for this model and show that it depends on the geometrical and topological properties of the fullerene.

Continuum framework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries

We present a continuum framework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries that are allowed to be nonplanar or nonequilibrium. In our continuum framework, we define a dislocation density potential function on the dislocation array surface or grain boundary to describe the orientation dependent continuous distribution of dislocations in a very simple and accurate way. The continuum formulations incorporate both the long-range dislocation interaction and the local dislocation line energy, and are derived from the discrete dislocation model. The continuum framework recovers the classical Read–Shockley energy formula when the long-range elastic fields of the low angle grain boundaries are canceled out. Applications of our continuum framework in this paper are focused on dislocation structures on static planar and nonplanar low angle grain boundaries and misfitting interfaces. We present two methods under our continuum framework for this purpose, including the method based on the Frank׳s formula and the energy minimization method. We show that for any (planar or nonplanar) low angle grain boundary, the Frank׳s formula holds if and only if the long-range stress field in the continuum model is canceled out, and it does not necessarily hold for a total energy minimum dislocation structure.

Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model

Cellular solids are usually treated as homogeneous continuums with effective properties. Nevertheless, these mechanical properties depend strongly on the ratio of the specimen size to the cell size. These size effects may be accounted for according to preliminary static analysis of effective continuums based on couple-stress theory. In this paper an effective dynamic continuum model, based on couple-stress theory, is proposed to analyze the behavior of free vibrations of periodic cellular solids. In this continuum model, the effective mechanical constants of the effective continuum are deduced by an equivalent energy method. The cellular solid structure is then replaced with the equivalent couple-stress continuum with same overall dimension and shape. Moreover, the finite element formulation of the couple-stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis. The eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method. A conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution. Several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model. Good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids.

Callan-Symanzik approach to infrared Yang-Mills theory

Dyson-Schwinger equations are the most common tool for the determination of the correlation functions of Landau gauge Yang-Mills theory in the continuum, in partic- ular in the infrared regime. We shall argue that the use of Callan-Symanzik renormaliza- tion group equations has distinctive advantages over the Dyson-Schwinger equations, in particular for the vertex functions. We present a generalization of the infrared safe renor- malization scheme proposed by Tissier and Wschebor in 2011. The comparison with the existing lattice data for the gluon and ghost propagators can be used to determine the most appropriate renormalization scheme. Dyson-Schwinger equations have given access to the deep infrared (IR) regime of Landau gauge Yang-Mills theory. The first solutions found show a scaling behavior of the propagators in the IR (1). Several years later, another type of solutions of the same equations was discovered (2), with a massive gluon propagator and a finitely enhanced ghost propagator in the IR (decoupling solutions). To date, Dyson-Schwinger equations are still the main tool for the (semi-)analytical exploration of the IR regime of Yang-Mills theory. In a parallel development that actually initiated much earlier with Gribov's observation of the existence of gauge copies in his famous 1978 paper (3) and was later worked out in great detail by Zwanziger (4), the theoretical foundations of IR Yang-Mills theory were laid. While this so-called Gribov-Zwanziger scenario seemed to confirm the existence of the scaling solutions, a more recent refinement (5) favors the decoupling type of solutions. Finally, the results of simulations of Yang- Mills theory in the Landau gauge on huge lattices (6) have been interpreted by most workers in the field as confirming the realization of the decoupling type of solutions in three and four space-time dimensions. In this contribution, we will present a different technique for the exploration of the IR regime of Yang-Mills theory. We intend to reproduce the results of lattice simulations in the Landau gauge that restrict the gauge field configurations to the (first) Gribov region, but make no attempt to reach the fundamental modular region. The restriction to the Gribov region implies the breaking of BRST invariance in the continuum formulation (4). The most important consequence of the broken BRST invariance for the IR regime is the appearance of a mass term for the gluon field (5). Indeed, the addition of a gluonic mass term to the Yang-Mills action has been shown to reproduce all the solutions (two types of scaling solutions and the decoupling solution) found before with the help of Dyson- Schwinger equations when solving the Callan-Symanzik equations for this theory in the IR regime in

Thermodynamic Formulation of a Material Model for Microcracking Applied to Creep Damage

A Hookean material containing an increasing number of non-interacting microcracks is studied from the aspect of continuum thermodynamics. Rectilinear microcracks in a two-dimensional body are examined. The former model by Basista (2003) is enhanced to describe the response of microcracks to compression. Microcrack densities introduced here enter into the formulation of continuum thermodynamics as internal variables. The strong physical foundation of these internal variables makes them more attractive quantities for damage mechanics than variable damage, the physical background of which is sometimes unclear. The model is coded as an Abaqus VUMAT subroutine and applied to the creep of ice.

Characterization of Microdamage Healing in Asphalt Concrete with a Smeared Continuum Damage Approach

The fatigue of asphalt concrete (AC) mixtures is a major source of distress in pavement structures. In most laboratory fatigue studies, testing is carried out by means of continuously repeating load cycles. However, under real traffic conditions, AC is subjected to a succession of load pulses as the traffic passes. Between these loads the material is not subjected to external forces and undergoes changes related to relaxation and healing, which together act to increase the overall fatigue life relative to what is observed under continuous loading. In the study reported in this paper, the effects of a rest period on the fatigue response in uniaxial, on-specimen, displacement-controlled loading was studied at 4.48C, 21.18C, and 388C. Loading consisted of a sinusoidal pulse–rest history with rest periods of 1, 5, or 10 s. Mixtures were used with the same aggregate structure and constituent materials but different asphalt and air void (AV) contents. It was found that the introduction of even short rest periods substantially improved the fatigue response of AC. A viscoelastic damage model for smeared healing was derived and characterized with the measured data. This model was compared with a damage-only continuum formulation and used to identify the impacts of healing, apart from any viscoelastic relaxation effects. It was found that greater AC and lower AV content yielded more favorable healing behaviors. The findings also suggested differences in the active healing mechanisms in pulse–rest and block–rest loading, typically used to study healing in AC. These differences are discussed.

배터리 전극 설계를 위한 응력-확산 완전연계 멀티스케일 해석기법

In this paper, we device stress-diffusion full coupling multiscale analysis method for battery electrode simulation. In proposed method, the diffusive and mechanical properties of electrode material depend on Li concentration are estimated using density function theory(DFT) simulation. Then, stress-diffusion full coupling continuum formulation based on finite element method(FEM) is constructed with the diffusive and mechanical properties calculated from DFT simulation. Finally, silicon nanowire anode charge and discharge simulations are performed using the proposed method. Through numerical examples, the stress-diffusion full coupling method shows more resonable results than previous one way continuum analysis.