Nonlinear Continuum Model Research Articles

A discontinuous Galerkin formulation for a strain gradient-dependent damage model

The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand C 1 continuity of the shape functions. Here, recent developments in discontinuous Galerkin methods are explored and exploited for the solution of a prototype non-linear strain gradient-dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard C 0 shape functions. The formulation is tested in one dimension for the simplest possible finite element formulation: continuous piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed formulation.

Localized defects in multilayer coatings

We present a non-linear continuum model of the growth of localized defects in multilayer coatings nucleated by particles on the substrate. The model is valid when the deposition and etch fluxes are near normal incidence so that shadowing effects are negligible. Three-dimensional simulations of defects in Mo/Si multilayer films nucleated by arrays of lithographically patterned particles are shown to be in good agreement with experimental measurements. Our results confirm that incorporating ion beam etching in the multilayer deposition process significantly suppresses the defect growth. This has a potentially important application in the fabrication of defect-free masks for extreme ultraviolet lithography.

Localized defects in multilayer coatings

We present a non-linear continuum model of the growth of localized defects in multilayer coatings nucleated by particles on the substrate. The model is valid when the deposition and etch fluxes are near normal incidence so that shadowing effects are negligible. Three-dimensional simulations of defects in Mo/Si multilayer films nucleated by arrays of lithographically patterned particles are shown to be in good agreement with experimental measurements. Our results confirm that incorporating ion beam etching in the multilayer deposition process significantly suppresses the defect growth. This has a potentially important application in the fabrication of defect-free masks for extreme ultraviolet lithography.

Dynamics of large-scale brain activity in normal arousal states and epileptic seizures.

Links between electroencephalograms (EEGs) and underlying aspects of neurophysiology and anatomy are poorly understood. Here a nonlinear continuum model of large-scale brain electrical activity is used to analyze arousal states and their stability and nonlinear dynamics for physiologically realistic parameters. A simple ordered arousal sequence in a reduced parameter space is inferred and found to be consistent with experimentally determined parameters of waking states. Instabilities arise at spectral peaks of the major clinically observed EEG rhythms-mainly slow wave, delta, theta, alpha, and sleep spindle-with each instability zone lying near its most common experimental precursor arousal states in the reduced space. Theta, alpha, and spindle instabilities evolve toward low-dimensional nonlinear limit cycles that correspond closely to EEGs of petit mal seizures for theta instability, and grand mal seizures for the other types. Nonlinear stimulus-induced entrainment and seizures are also seen, EEG spectra and potentials evoked by stimuli are reproduced, and numerous other points of experimental agreement are found. Inverse modeling enables physiological parameters underlying observed EEGs to be determined by a new, noninvasive route. This model thus provides a single, powerful framework for quantitative understanding of a wide variety of brain phenomena.

Microstructures and interfaces in Ni–Al martensite: comparing HRTEM observations with continuum theories

Different microstructural configurations of Ni–Al martensite are described on the meso- and atomic-scale, based on transmission electron microscopy images at conventional and atomic resolution. The observed structural features are described in close correlation with modern linear and non-linear continuum models, explaining the existence of particular orientations and stress accommodating features such as lattice reorientation, twin tapering and bending.

A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction

The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.

Nonlinear waves in thermo-magnetoelasticity II. Wave generation in a perfect electric conductor

Relying on a nonlinear continuum model of thermo-magnetoelasticity previously introduced by the authors, in which the heat capacity of the medium depends on strain and magnetic field, we investigate the effect of this dependence on the one-dimensional propagation of nonlinear, thermo-magnetoacoustic waves in an anisotropic (but centrosymmetric) half-space of a perfect electric conductor permeated by a constant magnetic field. The obtained formula for the relative variation of the phase velocity of the wave due to this dependence may be used, in combination with the proper experimental data, to predict the value of a material constant.

Monolayer instability: An upper bound for the wetting and nonwetting of a substrate by a solid monolayer.

Continuum mechanics and computer simulations are used to study the elastic stability of solid monolayers. We observe microscopically the upper limits of the ratio of adsorption to cohesive energies that allow for the wetting of the substrate by a solid monolayer using molecular dynamics. Also, structural instabilities are found analytically in nonlinear elastic continuum models by solving the nonlinear von K\'arm\'an equations for a thin circular system (an adsorbed island) undergoing planar compression. The instability occurs as a bifurcation at the first eigenvalue. A series of simulations over a range of adsorbate energies versus temperature demonstrates the existence of a stability boundary.

Nonlinear Model for Building‐Soil Systems

A finite-element based, numerical analysis methodology has been developed for the nonlinear analysis of building-soil systems. The methodology utilizes a reduced-order, nonlinear continuum model to represent the building, and the soil is represented with a simple nonlinear two-dimensional plane strain finite element. The foundation of the building is idealized as a rigid block and the interface between the soil and the foundation is modeled with an interface contract element. The objectives of the current paper are to provide the theoretical development of the system model, with particular emphasis on the modeling of the foundation-soil contact, and to demonstrate the special-purpose finite-element program that has been developed for nonlinear analysis of the building-soil system. Examples are included that compare the results obtained with the special-purpose program with the results of a general-purpose nonlinear finite-element program.

Incommensurate correlations in quantum spin models.

Frustrated quantum antiferromagnets in two dimensions are studied by analyzing, using a 1/N expansion, the phase properties of associated (2+1)-dimensional nonlinear continuum models. The spin system on square lattice with isotropic couplings that include up to the third-nearest neighbors is considered. Periodic, commensurate, or incommensurate correlations are found to be stable. The relevance in some phase regions of effective (1+1)-dimensional nonlinear models is discussed.

Development of a nonlinear continuum model for wave propagation in joined media: theory for single joint set

A method for constructing dispersive, nonlinear continuum models of jointed media is described herein and exemplified for the case of a single regular joint set. System nonlinearities in the example treated result from nonlinear joint properties. Model construction is based upon a homogenization technique which employs multivariable asymptotic expansions in conjunction with certain weighted residual procedures. The methodology furnishes the equations of motion, the appropriate initial and boundary conditions, and a set of consistent constitutive relations. For linear response, the derived model is validated by comparing the predicted phase velocity spectra with the exact spectra. For nonlinear response, model validation is accomplished by comparing predicted results for several transient wave propagation problems with “exact” data obtained by use of detailed FEM analyses wherein the microstructure of the joined media is modeled explicitly. The validation studies reveal that the derived continuum model provides good simulation of complex wave phenomena and furnishes an economical alternative to detailed, explicit FEM models. The studies performed reveal the importance of wave dispersion effects for nonlinear as well as linear joint responses. The research presented is in response to a need for an advanced computation model to be used in conjunction with explosive source identification.

Model for a decorated twin boundary in ferroelastic martensites.

The structure of static (11\ifmmode\bar\else\textasciimacron\fi{}0) solitary twin boundaries of cubic-tetragonal martensitic transformations in ferroelastic materials is calculated at all temperatures below the first-order transition temperature. These solutions also describe uniformly moving coherent (11\ifmmode\bar\else\textasciimacron\fi{}0) twin boundaries. These calculations are based on a nonlinear and nonlocal three-dimensional continuum model, with a two-component strain order parameter. The results relate to a special physical situation with a continuous distribution of dislocations decorating the twin boundary. Approaching the first-order transition temperature from below, the tetragonal-tetragonal solitary domain wall splits gradually into two cubic-tetragonal solitary domain walls of finite width. Their separation, however, diverges at the transition temperature. Above this temperature no stationary solutions are found, which may imply the possible existence of dynamic solutions with nontrivial (i.e., nontraveling-type) time dependence. Numerical experiments indicate that these dynamic solutions have a pulse shape and not a kink shape.

Nonlinear and nonlocal continuum model of transformation precursors in martensites

An intrinsic mechanism for explaining the origin of the transformation precursors observed above the transition in the parent phase of many materials undergoing a martensitic transformation is proposed. It is based on a nonlinear and nonlocal elastic continuum model for the elastic displacement field describing the parent-product deformation for the two-dimensional analog of a cubic-tetragonal transformation. By minimizing the Landau-Ginzburg free energy functional for the total elastic (strain plus strain gradient) energy a static, possibly stable, continuous, periodically modulated {110}/〈110〉 strain pattern is obtained which corresponds to alternating layers of more and of less transformed material, consistent with experimental observations. This pattern is stabilized by the balance between nonlinear and nonlocal elastic effects. Numerical application to In1-x XT x , alloys gives the minimum period of the modulation in the order of nanometers, in agreement with experimental observations.

Nonlinear Constitutive Relations for Human Brain Tissue

Head injury modeling studies warrant detailed investigations into the rheological behavior of brain tissue. This paper deals with the development of the concept of a power function and employs energy principles to derive nonlinear constitutive relations for human brain tissue. Nonlinear discrete and continuum models are constructed and it is found that the model predictions compare well with experimentally determined in-vitro properties of human brain tissue under uniaxial loading conditions.