Nonlocal Extension Research Articles

A micromechanics-based non-local anisotropic model for unilateral damage in brittle materials

The present Note is devoted to the formulation of a micromechanics-based model of non-local anisotropic damage and its application to concrete materials and structures. We first formulate a local anisotropic unilateral damage model on the basis of a suitable homogenization scheme which takes into account interactions between penny-shaped microcracks as well as their spatial distribution. The damage surface is built by using an energy release rate-based criterion. Then a non-local extension of the model is proposed by replacing the local energy release-rate for each family of microcracks by its average over a characteristic volume V of the material centered at a given point. In order to demonstrate the efficiency of the non-local model in mesh-independent simulation of failure process in structures, some applications concerning failure of concrete materials and structures are presented. To cite this article: Q.-z. Zhu et al., C. R. Mecanique 336 (2008).

Quark model description of quasi-elastic pion knockout from the proton at JLAB

The interference term between s- and t-pole contributions to the p(e,e′π+)n cross section is evaluated on the basis of the constituent quark model. It is shown that the contribution of baryon s-poles can be modeled by a nonlocal extension of the Kroll–Ruderman contact term. This contribution is in a destructive interference with the pion t-pole that is essential to improve the description of recent JLab data at the invariant mass W=1.95 GeV. Some predictions are made for a new JLab measurement at higher values W=2.1–2.3 GeV and Q2 centered at 1.6 and 2.45 GeV2/c2.

Weakly non-local fluid mechanics: the Schrödinger equation

A weakly non-local extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit, the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of the Schrödinger equation (stochastic, Fisher information, exact uncertainty) is clarified.

Incompatibility Induced Nonlocal Crystal Plasticity

AbstractMotivated by the fact that locally inhomogeneous elastic or plastic deformations may result in incompatibilities of the fictitious intermediate configuration a strain gradient crystal plasticity model is developed. Thereby incompatibilities can be accounted for and scale dependent material behavior, as also observed experimentally, is predictable. A nonlocal extension of existing local formulations is proposed which does not require additional boundary conditions and thus maintains the classical BVP structure. On the numerical side key developments are an extended FE‐formulation for rate‐(in)dependent strain gradient plasticity and a local FE‐formulation which bases the gradient computation on an operator split combined with a smoothing algorithm. Comparative numerical studies for classical examples proove the superior efficiency of the second approach. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical simulations of localization of deformation in quasi-brittle materials within non-local softening plasticity

The paper presents results of FE-calculations on shear localizations in quasi-brittle materials during both an uniaxial plane strain compression and uniaxial plane strain extension. An elasto-plastic model with a linear Drucker-Prager type criterion using isotropic hardening and softening and non-associated flow rule was used. A non-local extension was applied in a softening regime to capture realistically shear localization and to obtain a well-posed boundary value problem. A characteristic length was incorporated via a weighting function. Attention was focused on the effect of mesh size, mesh alignment, non-local parameter and imperfections on the thickness and inclination of shear localization. Different methods to calculate plastic strain rates were carefully discussed.

Microstructure evolution in the equal channel angular extrusion process

We apply a theory of single-crystal plasticity with microstructure to the simulation of the ECAE process. The specific microstructures considered in the simulations are of the sequential lamination type. The size of the microstructure is estimated a posteriori by means of a nonlocal extension of the theory which accounts for dislocation energies. Texture evolution is calculated simply by recourse to Taylor’s hypothesis. Calculations concerned with an FCC material (Al–Cu alloy) and 90° ECAE reveal a wealth of information regarding the geometry, size, and texture evolution of subgrain microstructures. The predicted sizes and textures are in good quantitative agreement with the available experimental data.

A numerical model to describe damage in glass under impact loading

This work presents an anisotropic damage model and its non-local extension to deal with fragmentation induced by impact loadings. Discrete cracks are described via non-local damage parameters. The damage model is based on quasi-static properties of flaw distributions and enables us lo reproduce dynamic cracking patterns obtained in edge-on-impact tests. Different regimes such as single on multiple fragmentation appear and can be explamed by this approach.

Local and non-local Gurson-based ductile damage and failure modelling at large deformation

Abstract The purpose of this work is the formulation, numerical implementation and initial application of a non-local extension of existing Gurson-based modelling for isotropic ductile damage and attendant crack growth. It is being carried out under the premise that void coalescence results not only in accelerated damage development (e.g., Needleman and Tvergaard, 1984), but also in damage delocalisation (i.e., via interaction between neighbouring Gurson RVE's). To this end, we proceed by analogy with the approach of Needleman and Tvergaard (1984) who replaced the Gurson void volume fraction f with a (local) effective damage parameter f ∗ in the Gurson yield condition to account for the effect of void coalescence on the material behaviour. In the current case, the role of f ∗ is taken over and generalised by an effective continuum damage field ν. A field relation for ν is formulated here in the framework of continuum thermodynamics. In the simplest case, the resulting relation is formally analogous to the inhomogeneous temperature equation in which void nucleation and growth represent (local) sources for ν and in which void coalescence takes place in a process zone whose dimension is determined by a characteristic material lengthscale. Analogous to temperature, then, ν represents an additional continuum degree-of-freedom here, resulting in a coupled deformation-damage field model. In the last part of the work, the complete model for coupled damage-deformation is implemented numerically using the finite-element method on the basis of backward-Euler integration and consistent linearisation. Using this implementation, the behaviour of the current extended Gurson-based damage model is investigated for the case of simple tension of an inhomogeneous steel block. In particular, the corresponding simulation results document quantitatively the dependence of the delocalisation of the model damage process and minimisation of mesh-dependence on the characteristic dimension of the damage process zone.

A non-local extension of Gurson-based ductile damage modeling

The purpose of this work is the development and numerical implementation of a non-local extension of existing Gurson-based modeling for isotropic ductile damage and attendant crack growth. In the current work, this extension is based on a generalization of Gurson-based ductile damage modeling formally analogous to that of Needleman and Tvergaard as based on a (local) effective damage parameter f * which accounts in their case in an effective fashion for the effect of void coalescence on yield behaviour. Here, the corresponding generalized effective damage parameter ν is introduced in order to delocalize the model damage process. To this end, ν is modeled in this work as a scalar-valued continuum microstructural field or generalized phase field via a recent thermodynamic approach to the modeling of such fields. In the last part of the work, the complete model for coupled damage-deformation is implemented numerically using the finite element method and utilized to investigate the damage behaviour of an inhomogeneous steel block in simple tension. The corresponding simulation results demonstrate in particular that delocalization of the model damage process also leads to minimization of mesh-dependence. For simplicity, the current formulation is restricted to the case of small strain and isothermal conditions.

Theory of phase separation kinetics in polymer–liquid crystal systems

We introduce a kinetic model describing the phase separation in the mixture of long rod-like molecules and long chain-like molecules. The model uses the angular distribution function for the orientations of the rods as a dynamical variable. The energetics is based on the nonlocal Onsager theory for the rods combined with a nonlocal extension of the Flory–Huggins theory. The kinetics explicitly takes into account the preferential diffusion along the rods. We computed the phase diagrams in this model and found a number of transitions leading to phase separation. We also performed numerical simulations of the phase separation kinetics and studied the resulting morphologies.

Magnetic moment and radius of the nucleon in a nonlocal model of meson-nucleon interaction

Magnetic moment and radius of the nucleon are calculated in a nonlocal extension of the chiral linear \sigma-model. Properties of the nonlocal model under the vector and axial transformations are considered. The conserved electromagnetic and vector currents, and partially conserved axial vector current are obtained. In the calculation of the nucleon electromagnetic vertex the \pi- and \sigma-loop diagrams are included. Contribution from vector mesons is added in the vector meson dominance model with a gauge-invariant photon-meson coupling. The nonlocality parameter associated with the \pi N interaction is fixed from the experimental magnetic moment of the neutron. Other parameters (nonlocality parameter for the \sigma N interaction and the mass of the \sigma-meson) are constrained by the magnetic moment of the proton. The calculated electric and magnetic mean-square radii of the proton and neutron are in satisfactory agreement with experiment.

Chiral phase transition in a covariant nonlocal NJL model

The properties of the chiral phase transition at finite temperature and chemical potential are investigated within a nonlocal covariant extension of the Nambu–Jona-Lasinio model based on a separable quark–quark interaction. We consider both the situation in which the Minkowski quark propagator has poles at real energies and the case where only complex poles appear. In the literature, the latter has been proposed as a realization of confinement. In both cases, the behaviour of the physical quantities as functions of T and μ is found to be quite similar. In particular, for low values of T the chiral transition is always of first order and, for finite quark masses, at certain “end point” the transition turns into a smooth crossover. In the chiral limit, this “end point” becomes a “tricritical” point. Our predictions for the position of these points are similar, although somewhat smaller, than previous estimates. Finally, the relation between the deconfining transition and chiral restoration is also discussed.

Wick Power Series Converging to Nonlocal Fields

We consider the infinite series in Wick powers of a generalized free field that are convergent under smoothing with analytic test functions and realize a nonlocal extension of the Borchers equivalence classes. The nonlocal fields to which the Wick power series converge are proved to be asymptotically commuting. This property serves as a natural generalization of the relative locality of the Wick polynomials. The proposed proof is based on exploiting the analytic properties of the vacuum expectation values in the x space and applying the Cauchy–Poincare theorem.

An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions

We study the existence and stability of stationary solutions of an integrodifferential model for phase transitions, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. As such, this model is a nonlocal extension of the AllenCahn equation, which incorporates long-range interactions. We find that the set of stationary solutions for this model is much larger than that of the AllenCahn equation.

Equivalent bosonic theory for the massive Thirring model with a non-local interaction

We study, through path-integral methods, an extension of the massive Thirring model in which the interaction between currents is non-local. By examining the mass expansion of the partition function we show that this non-local massive Thirring model is equivalent to a certain non-local extension of the sine-Gordon theory. Thus, we establish a non-local generalization of the well known Coleman's equivalence. We also discuss some possible applications of this result in the context of one-dimensional strongly correlated systems and finite-size quantum field theories.

Anisotropic Upper Critical Field of LuNi2B2C

The upper critical field, ${H}_{c2}$, of LuN${\mathrm{i}}_{2}$${\mathrm{B}}_{2}$C has been determined from the temperature dependence of the magnetization. A temperature dependent anisotropy within the basal plane which decreases from a value of 1.1 at 4.5 K to a value of 1.0 at ${T}_{c}$ and an almost temperature independent out-of-plane anisotropy have been observed. Near ${T}_{c}$ the upper critical field along all directions shows a strong upward curvature. These features of ${H}_{c2}$ can be explained quantitatively using a nonlocal extension of the Ginzburg-Landau equations. The Fermi velocity averages determined from these measurements and those obtained in band structure calculations are in good agreement.

Meson masses within the model of induced nonlocal quark currents.

The model of induced quark currents formulated in our recent paper (Phys. Rev. D51, 176) is developed. The model being a kind of nonlocal extension of the bosonization procedure is based on the hypothesis that the QCD vacuum is realized by the (anti-)self-dual homogeneous gluon field. This vacuum field provides the analytical quark confinement. It is shown that a particular form of nonlocality of the quark and gluon propagators determined by the vacuum field, an interaction of quark spin with the vacuum gluon field and a localization of meson field at the center of masses of two quarks can explain the distinctive features of meson spectrum: Regge trajectories of radial and orbital excitations, mass splitting between pseudoscalar and vector mesons, the asymptotic mass formulas in the heavy quark limit: $M_{Q\bar Q}\to 2m_Q$ for quarkonia and $M_{Q\bar q}\to m_Q$ for heavy-light mesons. With a minimal set of parameters (quark masses, vacuum field strength and the quark-gluon coupling constant) the model describes to within ten percent inaccuracy the masses and weak decay constants of mesons from all qualitatively different regions of the spectrum.

On purely elsatic instabilities in eccentric cylinder flows

An experimental and theoretical investigation of the stability of the viscoelastic flow of the Boger fluid between eccentric cylinders is considered. In our theoretical study, a local linear stability analysis for the flow of an Oldroyd-B fluid suggests that the flow is elastically unstable for all eccentricities. It is shown that the linear stability of eccentric cylinder flow, to leading order for small gap ratio, is very similar to that of Taylor-Dean flow (Y.L. Joo and E.S.G. Shaqfeh, Phys. Fluids A, 4(3) (1992) 524). Flow visualization experiments with a solution of high molecular polyisobutylene dissolved in a viscous solvent clearly show the transition from a purely azimuthal to a secondary torodial flow. Comparison of these experimental results with the local linear stability theory shows qualitative agreement with the predicted critical conditions for the onset of the non-inertial cellular instability when δ < 0.4, where δ is the eccentricity (made dimensionless with the average gap thickness). At higher eccentricities, experiment and theory cease to agree. Evidence is given that this disagreement is due to the non-local streamwise extension rate which is not included in the present theory, and whose magnitude increases with increasing δ. We look forward to a comprehensive investigation of this interesting phenomenon.

Group interpretation of the spectral parameter in the case of nonhomogeneous, nonlinear Schrödinger system

A one-parameter group inserting the spectral parameter into the nonparametric Lax pair of the nonhomogeneous, nonlinear Schrödinger system is presented and discussed. Being a nonlocal extension of a Lie point transformation this group is probably not contained in any known class of symmetries.

NEW METHODS IN QUANTUM NON-LOCAL FIELD THEORY

Functional methods are developed which serve to simplify greatly the calculations in quantum non-local field theory (QNFT). The techniques also serve to give an insight into the underlying structure of QNFT. We show that a transformation can be defined which relates the QNFT Lagrangian to its local antecedent. We prove that the non-local extension of the local gauge symmetry can be obtained by applying this transformation to the local gauge transformation. The utility of this method is demonstrated by an explicit application to both scalar electrodynamics and Yang-Mills field theory.