The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture

We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain.     

A P-matrix-based model for the analysis of SAW transversely coupled resonator filters, including guided modes and a continuum of radiated waves

When designing transversely coupled resonator filters, unexpected spurii are often observed on the high-frequency side of the transfer function. These spurii cannot be described using only the classical waveguide model. Discrete transverse modes inside the grating can be identified if one assumes that the modes have exponential decay outside the grating; however, a continuum of solutions exist in the case of propagating waves outside the grating. A large part of the source excitation may be coupled to these radiated waves. To include this phenomena in the model, a decomposition on the above mentioned continuum was performed. We describe our P-matrix-based model for transversely coupled structures. This model takes into account all guided modes and the continuum. It allows the use of an arbitrary number of acoustical layers and electrical ports. A comparison of the measured and simulated frequency responses is presented for different filters and different metallization thickness showing an excellent agreement.     

Concurrently coupled atomistic and XFEM models for dislocations and cracks

A method for the modeling of dislocations and cracks by atomistic/continuum models is described. The methodology combines the extended finite element method with the bridging domain method (BDM). The former is used to model crack surfaces and slip planes in the continuum, whereas the BDM is used to link the atomistic models with the continuum. The BDM is an overlapping domain decomposition method in which the atomistic and continuum energies are blended so that their contributions decay to their boundaries on the overlapping subdomain. Compatibility between the continua and atomistic domains is enforced by a continuous Lagrange multiplier field. The methodology allows for simulations with atomistic resolution near crack fronts and dislocation cores while retaining a continuum model in the remaining part of the domain and so a large reduction in the number of atoms is possible. It is applied to the modeling of cracks and dislocations in graphene sheets. Energies and energy distributions compare very well with direct numerical simulations by strictly atomistic models. Copyright © 2008 John Wiley & Sons, Ltd.     

An analysis of dynamic fracture in concrete with a continuum visco-elastic visco-plastic damage model

We present a rate-dependent continuum model for the dynamic modelling of concrete. Rate dependency is included by means of visco-elastic and visco-plastic constitutive relationships. The inclusion of the Stefan effect, described by a visco-elastic model, results in an increased tensile strength with increasing loading rate. An additional rate effect comes from the visco-plastic contribution which, from a mechanical standpoint, can be related to the micro-inertia of the material surrounding the crack tip. Static and dynamic examples show the capabilities of the model with respect to localization, hysteresis and dissipation of energy.     

Mass transport through swelling membranes

This paper deals with non-Fickian mass transport through polymeric membranes. The process is described via continuum mechanics. We introduced an appropriate function relating the stress to concentration and time, such that the model predicts a realistic stress distribution at equilibrium also. The effect of different dimensionless groups is illustrated and quantitative agreement with experimental data is shown for transport of organic solvents through PVC.     

SO(3) nonlinear σ model for a spiral phase in the spin-fermion model

Starting from a doped spin-fermion model for the CuO₂ planes in high-T_c superconductors, we derive an effective continuum theory for the spin degrees of freedom by means of a gradient expansion. We assume an incommmensurate, spiral configuration for the spins. Extending our previous treatment of a planar spiral, we allow for three-dimensional fluctuations of the spin fields, which are described by an S0(3) matrix order parameter. The continuum limit is obtained by a systematic expansion in powers of a short distance cutoff. The occurring infinite series is summed to all orders using a combinatorial method which exploits the constraint obeyed by the S0(3) order parameter. The resulting continuum theory is given by an S0(3) quantum nonlinear model, where the influence of doping is contained in the fermionic susceptibilities that enter into the coupling constants of the model.     

Introduction to computational models of damage dynamics under stochastic actionst

This paper reviews and discusses some basic ingredients necessary for the study of damaged continua with diffused defects like microcracks, pores, dislocations, etc., under stochastic loading histories and, in particular, under sequences of impulses described by Poisson arrival processes. The mechanical model of a continuum with microstructure is adopted: in other words, the state of the continuum is described by the usual displacement field and by an additional field of a second-order non-symmetric tensor which describes the microstructural rearrangement of the material due to the presence of defects. It is shown that the time evolution of this tensor, usually assumed empirically on the basis of experimental results, is governed by a balance equation. The discretization of the problem and integral measures of damage, useful for the numerical solutions, are also discussed.     

Volume defects in nonlocal micropolar elasticity

Using the concept of prescribed arrays of point forces and point couples, the long range displacement, the stress and the interaction energy fields associated with an isolated volume defect in nonlocal micropolar continuum elasticity are calculated. It is shown that all the physically possible volume defects can be described by a distribution of body forces and body couples. Interestingly these forces and couples are not singular extending therefore the validity of the classical continuum treatment of defects. Line defects such as dislocations and disclinations can be analyzed in a similar way.     

Elastic continuum theories of lattice defects: a review

The presently available elastic continuum theories of lattice defects are reviewed. After introducing a few elementary concepts and the basic equations of elasticity the Eshelby’s theory of misfitting inclusions and inhomogeneities is outlined. Kovács’ result that any lattice defect can be described by a surface distribution of elastic dipoles is described. The generalization of the isotropic continuum approach to anisotropic models and to Eringen’s isotropic but non-local model is discussed. Kröner’s theroy (where a defect is viewed as a lack of strain compatibility in the medium) and the elastic field equations (formulated in a way analogous to Maxwell’s field equations of magnetostatics) are described. The concept of the dislocation density tensor is introduced and the utility of higher-order dislocation density correlation tensors is discussed. The beautiful theory of the affine differential geometry of stationary lattice defects developed by Kondo and Kröner is outlined. Kosevich’s attempt to include dynamics in the elastic field equations is described. Wadati’s quantum field theory of extended objects is mentioned qualitatively. Some potential areas of research are identified.     

A robust nano-mechanics approach for tensile and modal analysis using atomistic–continuum mechanics method

Nanoscale engineering has been developing rapidly. However, experimental investigations at the nanoscale level are very difficult to conduct. This research seeks to employ the same model to investigate an atomic-scale structure for tensile and modal analyses, based on atomistic–continuum mechanics (ACM) and a finite element method (FEM). The ACM transfers an originally discrete atomic structure into an equilibrium continuum model using atomistic–continuum transfer elements. All interatomic forces, described by the empirical potential functions, can be transferred into springs to form the atomic structure. The spring network models were also widely utilized in FEM based nano-structure studies. Thus, this paper attempts to explore ACM using three examples including silicon, carbon nanotube, and copper. All of the results are validated by bulk properties or literature.     

Continuum kinematical modeling of mass increasing biological growth

Constructing continuum kinematical models of mass increasing biological growth has been the objective of many studies in the last 100 years. The significant features of the successful studies are briefly described, critically reviewed and organized in the contemporary notation of continuum kinematics. While success has been achieved in many kinematic tissue modeling categories, an agreed upon kinematic model for the large deformations of soft tissue remains a challenge.     

Simulations of continuum coherent states and its use in quantum cryptographic systems

Abstract

The continuum states formalism is suitable for field quantization in optical fibre; however, they are harder to use than discrete states. On the other hand, a Hermitian phase operator can be defined only in a finite dimensional space. We approximated a coherent continuum state by a finite tensor product of coherent states, each one defined in a finite dimensional space. Using this, in the correct limit, we were able to obtain some statistical properties of the photon number and phase of the continuum coherent states from the probability density functions of the individual, finite dimensional, coherent states. Then, we performed a simulation of the BB84 protocol, using the continuum coherent states, in a fibre interferometer commonly used in quantum cryptography. We observed the fluctuations of the mean photon number in the pulses that arrive at Bob, which occurs in the practical system, introduced by the statistical property of the simulation.     

Size effects in the mechanical behavior of cellular materials

Effective mechanical properties of cellular materials depend strongly on the specimen size to the cell size ratio. Experimental studies performed on aluminium foams show that under uniaxial compression, the stiffness of these materials falls below the corresponding bulk value, when the ratio of the specimen size to the cell size is small. Conversely, in the case of simple shear and indentation, the overall stiffness rises above the bulk value. Classical continuum theory, lacking a length scale, cannot explain this size dependent mechanical behaviour. One way to account for these size effects is to explicitly model the discrete cellular morphology. We performed shear, compression and bending tests using discrete models, for hexagonal (regular and irregular) microstructures. Even though discrete models give a very good agreement with the experiments, they are computationally expensive for complex microstructures, especially in three dimensions. To overcome this, one can use a generalized continuum theory, such as Cosserat continuum theory, which incorporates a material length scale. We fit the Cosserat elastic constants of the models by comparing the discrete calculations with the analytical Cosserat continuum solutions in terms of macroscopic properties. We critically address the limitations of the Cosserat continuum theory.     

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Discrete and continuum methods for numerical simulations of non‐linear wave propagation in discontinuous media

The main objective of this paper is to develop a methodology to model dynamic loading of various discontinuous media. Two methods for modeling non‐linear waves in a media with multiple discontinuities are considered. The first one, a discrete method, is based on the Simple Common Plane contact algorithm. This method can be applied both to compliant contacts characterized by finite thickness and elastic moduli (such as joints in geomechanics) as well as to non‐compliant frictional contacts traditionally described by the slide lines in finite element/finite difference codes. The second one, a continuum method, assumes that the contacts are not compliant and can be modeled as one or several weakness planes cutting through the elements of the computational mesh. Both discrete and continuum methods described in the paper can be applied to derive equivalent continuum properties for media with multiple discontinuities. An example of such application for a randomly jointed media is given in the paper. Copyright © 2010 John Wiley & Sons, Ltd.     

Stable supercontinuum generation in short lengths of conventional dispersion-shifted fiber

By propagating 500-fs pulses through 2.5 m of standard fiber followed by 2 m of dispersion-shifted fiber, we generated >200 nm of spectral continuum between 1430 and 1630 nm, which is flat to less than ?0.5 dB over more than 60 nm. Pulses obtained by filtering the continuum show no increase in timing jitter over the source laser and are pedestal free to >28 dB, indicating excellent stability and coherence. We show that the second- and third-order dispersions of the continuum fiber and self-phase modulation are primarily responsible for the continuum generation and spectral shaping and found close agreement between simulations and experiments.