Continuum Formulation Research Articles

Characterization of fracture processes by continuum and discrete modelling

A large number of methods to describe fracture mechanical features of structures on basis of computational algorithms have been developed in the past due to the importance of the topic. In this paper, current and promising numerical approaches for the characterization of fracture processes are presented. A fracture phenomenon can either be depicted by a continuum formulation or a discrete notch. Thus, starting point of the description is a micromechanically motivated formulation for the development of a local failure situation. A current, generalized method without any restriction to material modelling and loading situation in order to describe an existing crack in a structure is available through the material force approach. One possible strategy to simulate arbitrary crack growth is based on an adaptive implementation of cohesive elements in combination with the standard discretization of the body. In this case, crack growth criteria and the determination of the crack propagation direction in combination with the modification of the finite element mesh are required. The nonlinear structural behaviour of a fibre reinforced composite material is based on the heterogeneous microstructure. A two-scale simulation is therefore an appropriate and effective way to take into account the scale differences of macroscopic structures with microscopic elements. In addition, fracture mechanical structural properties are far from being sharp and deterministic. Moreover, a wide range of uncertainties influence the ultimate load bearing behaviour. Therefore, it is evident that the deterministic modelling has to be expanded by a characterization of the uncertainty in order to achieve a reliable and realistic simulation result. The employed methods are illustrated by numerical examples.

Instabilities and oscillations in isotropic active gels

We present a generic formulation of the continuum elasticity of an isotropic crosslinked active gel. The gel is described by a two-component model consisting of an elastic network coupled frictionally to a permeating fluid. Activity is induced by active crosslinkers that undergo an ATP-activated cycle and transmit forces to the network. The on/off dynamics of the active crosslinkers is described via rate equations for unbound and bound motors. For large activity motors yield a contractile instability of the network. At smaller values of activity, the on/off motor dynamics provides an effective inertial drag on the network that opposes elastic restoring forces, resulting in spontaneous oscillations. Our work provides a continuum formulation that unifies earlier microscopic models of oscillations in muscle sarcomeres and a generic framework for the description of the large scale properties of isotropic active solids.

Lagrangian formulation of continuum with internal long-range interactions

Based on a new definition of nonlocal variable, this paper establishes the Lagrangian formulation for continuum with internal long-range interactions. Distinguished from the existing theories, the nonlocal term in the Lagrangian formulation automatically satisfies the zero mean condition determined by the action and reaction law. By this formulation, elastic wave in a rod with the internal long-range interactions is investigated. The dispersion of the elastic wave is predicted.

Continuum modeling of van der Waals interactions between carbon onion layers

A continuum formulation of the van der Waals interactions between carbon onion layers in form of pressure/radius relations is obtained. The relations are derived analytically considering the doubly-curved geometry of carbon onion layers. For finite layer radii it is shown that the van der Waals induced pressures on opposing faces of two adjacent onion layers are not equal and depend on both layer radii. Equilibrium configurations of different double-layer onion models are calculated using the theorem of stationarity of the total potential energy. The results are compared with those of simplified models which do not consider any curvature effects. It is shown that curvature effects have no significant influence on the equilibrium interlayer distances, but they significantly change the results for the radial displacements and, consequently, for the membrane forces in the layers. Comparison with Monte Carlo simulations of C60 in C180 and C60 in C240 shows that the derived van der Waals model represents the radial displacements and equilibrium interlayer distances better than the simplified models. Thus, whenever a good representation of the membrane forces is important, e.g., in stability analysis, the new approach should be used.

Computational continua

AbstractWe develop a new coarse‐scale continuum formulation hereafter referred to as computational continua. By this approach, the coarse‐scale governing equations are stated on a so‐called computational continua domain consisting of disjoint union of computational unit cells, positions of which are determined to reproduce the weak form of the governing equations on the fine scale. The label ‘computational’ is conceived from both theoretical and computational considerations. From a theoretical point of view, the computational continua is endowed with fine‐scale details; it introduces no scale separation; and makes no assumption about infinitesimality of the fine‐scale structure. From computational point of view, the computational continua does not require higher‐order continuity; introduces no new degrees‐of‐freedom; and is free of higher‐order boundary conditions. The proposed continuum description features two building blocks: the nonlocal quadrature scheme and the coarse‐scale stress function. The nonlocal quadrature scheme, which replaces the classical Gauss (local) quadrature, allows for nonlocal interactions to extend over finite neighborhoods and thus introduces nonlocality into the two‐scale integrals employed in various homogenization theories. The coarse‐scale stress function, which replaces the classical notion of coarse‐scale stress being the average of fine‐scale stresses, is constructed to express the governing equations in terms of coarse‐scale fields only. Perhaps the most interesting finding of the present manuscript is that the coarse‐scale continuum description that is consistent with an underlying fine‐scale description depends on both the coarse‐scale discretization and fine‐scale details. As a prelude to introducing the computational continua framework, we unveil the relation between the generalized continua and higher‐order mathematical homogenization theory and point out to their limitations. This serves as motivation to the main part of the manuscript, which is the computational continua formulation. Copyright © 2010 John Wiley & Sons, Ltd.

Flutter of an Arch Bridge via a Fully Nonlinear Continuum Formulation

A fully nonlinear parametric model for wind-excited arch bridges is proposed to carry out the flutter analysis of Ponte della Musica under construction in Rome. Within the context of an exact kinematic formulation, all of the deformation modes are considered (extensional, shear, torsional, in-plane, and out-of-plane bending modes) both in the deck and supporting arches. The nonlinear equations of motion are obtained via a total Lagrangian formulation while linearly elastic constitutive equations are adopted for all structural members. The parametric nonlinear model is employed to investigate the bridge limit states appearing either as a divergence bifurcation (limit point obtained by path following the response under an increasing multiplier of the vertical accidental loads) or as a Hopf bifurcation of a suitable eigenvalue problem (where the bifurcation parameter is the wind speed). The eigenvalue problem ensues from the governing equations of motion linearized about the in-service prestressed bridge con...

A Generalized Continuum Formulation for Composite Microcracked Materials and Wave Propagation in a Bar

A multifield continuum is adopted to grossly describe the dynamical behavior of composite microcracked solids. The constitutive relations for the internal and external (inertial) actions are obtained using a multiscale modeling based on the hypotheses of the classical molecular theory of elasticity and the ensuing overall elastodynamic properties allow us to take properly into account the microscopic features of these materials. Referring to a one-dimensional microcracked bar, the ability of such a continuum to reveal the presence of internal heterogeneities is investigated by analyzing the relevant dispersive wave propagation properties. Scattering of traveling waves is shown to be associated with the microcrack density in the bar.

A micromechanically motivated higher‐order continuum formulation of linear thermal conduction

AbstractA higher‐order continuum theory of linearized thermal conduction is developed where a rigid heat conductor is modeled as a material of grade‐2 with a view towards capturing the thermal effects in BGK‐Burnett type formulations for microfluidic flows. The construction is based on a second‐order thermal homogenization framework which leads to the identification of a thermal dissipation potential that is a function of the higher‐order gradients of the temperature. The dissipation potential delivers the expressions for the higher‐order fluxes and forms the basis of a discussion regarding the satisfaction of the second law of thermodynamics. Further restrictions on the constitutive expressions arise from material symmetry considerations. Strong and weak formulations of the associated boundary value problem are derived based on an appropriate identification of the boundary conditions. The methodologies employed and the implications of the theory are reminiscent of similar approaches in linearized elasticity formulations and are compared with micromorphic and Cahn‐Hilliard type formulations.

The exact decomposition of gauge variables in lattice Yang–Mills theory

In this Letter, we consider lattice versions of the decomposition of the Yang–Mills field a la Cho–Faddeev–Niemi, which was extended by Kondo, Shinohara and Murakami in the continuum formulation. For the SU(N) gauge group, we propose a set of defining equations for specifying the decomposition of the gauge link variable and solve them exactly without using the ansatz adopted in the previous studies for SU(2) and SU(3). As a result, we obtain the general form of the decomposition for SU(N) gauge link variables and confirm the previous results obtained for SU(2) and SU(3).

Modeling crowd dynamics by the mean-field limit approach

We propose to couple through optimal control theory, a dynamical crowd evolution models where pedestrians make decisions with a well-defined set of actions and strategies by minimizing their utility function. We explain how situations of this kind can be modeled by making use of differential game theory. By leveraging on some simple examples, we introduce the most fundamental concept of mean field games recently introduced by Lasry and Lions [J.-M. Lasry, P.-L. Lions, Mean field games, Jpn. J. Math. 2 (1) (2007) 229–260]. The main interest of mean field games in our work is to simplify the interactions between pedestrians. As a consequence, we derive a continuum formulation of the crowd dynamics and nonlinear systems involving partial differential equations of crowd models.

A Coupled Model of Two-Phase Diffusion and Flow through Deforming Porous Cosserat Media

Simulation of mining induced rock deformation, rock fracture enhanced permeability and fluid and gas diffusion and flow process is a complex task. A new three dimensional coupled mechanical two-phase double porosity desorption and diffusion finite element code called COSFLOW has been recently developed by CSIRO Exploration and Mining to service the mining industry’s need. A unique feature of COSFLOW is the incorporation of Cosserat continuum theory in its formulation. In the Cosserat model, inter-layer interfaces (joints, bedding planes) are considered to be smeared across the mass, i.e. the effects of interfaces are incorporated implicitly in the choice of stress-strain model formulation. An important feature of the Cosserat model is that it incorporates bending rigidity of individual layers in its formulation and this makes it different from other conventional implicit models. The Cosserat continuum formulation has a major advantage over conventional continuum models in that it can efficiently simulate rock breakage and slip as well as separation along the bedding planes. Any opening/closure along a bedding plane may introduce a strong anisotropy in fluid flow properties of the porous medium. This, in turn, will impact on the fluid/gas flow behaviour of the porous medium. This paper will briefly describe the Cosserat continuum theory, the treatment of permeability changes with rock deformation and the coupling of the two-phase dual porosity fluid diffusion- flow model and present a number of examples highlighting the capability of the developed code in simulating the mining induced rock deformation, permeability changes and fluid diffusion and flow will be presented.

Professor Ioannis Vardoulakis (1949–2009)

Professor Ioannis Vardoulakis (1949–2009)

An implicit stress gradient plasticity model for describing mechanical behavior of planar fiber networks on a macroscopic scale

The plasticity behavior of fiber networks is governed by complex mechanisms. This study examines the effect of microstructure on the macroscopic plastic behavior of two-dimensional random fiber networks such as strong-bonded paper. Remote load is a pure macroscopic mode I opening field, applied via a boundary layer assuming small scale yielding on the macroscopic scale. It is shown that using a macroscopic classical homogeneous continuum approach to describe plasticity effects due to (macroscopic) singular-dominated strain fields in planar fiber networks leads to erroneous results. The classical continuum description is too simple to capture the essential mechanical behavior of a network material since a structural effect, that alters the macroscopic stress field, becomes pronounced and introduces long-ranging microstructural effects that have to be accounted for. Because of this, it is necessary to include a nonlocal theory that bridges the gap between microscopic and macroscopic scales to describe the material response in homogeneous continuum models. An implicit stress gradient small deformation plasticity model, which is based on a strong nonlocal continuum formulation, is presented here that has the potential to describe the plasticity behavior of fiber networks on a macroscopic scale. The theory is derived by including nonlocal stress terms in the classical associated J2-theory of plasticity. The nonlocal stress tensor is found by scaling the local Cauchy stress tensor by the ratio of nonlocal and local von Mises equivalent stresses. The model is relatively easy to implement in ordinary finite element algorithms for small deformation theory. Fairly good agreements are obtained between discrete micromechanical network models and the derived homogeneous nonlocal continuum model.

A formulation for the micromorphic continuum at finite inelastic strains

The paper presents a generalized theory of deformation which can capture scale effects also in a homogenously deforming body. Scale effects are relevant when the dimensions of the specimen or structure themselves are in the micron and submicron scale, but also when it comes to high strain concentrations as in the case of localised shear bands or at crack tips, etc. In this context, so-called generalized continuum formulations have been proven to provide remedy as they allow for the incorporation of internal length-scale parameters which reflect the micro-structural influence on the macroscopic material response. Here, we want to adopt a generalized continuum framework which is based on the mathematical description of a combined macro- and micro-space ( Sansour, 1998b). The approach introduces additional degrees of freedom which constitute a so-called micromorphic deformation. First the treatment presented is general in nature but will be specified for the sake of an example and the number of extra degrees of freedom will be reduced to four. Based on the generalized deformation description new strain and stress measures are defined which lead to the formulation of a corresponding generalized variational principle. The theory is completed by Dirichlet boundary conditions for the displacement field. Of great advantage is the fact that the constitutive law is defined in the generalized space but can be classical otherwise. This limits the number of the extra material parameters necessary to those needed for the specification of the micro-space, in the example presented to only one. An example of scale effects in a homogenously deforming specimen and a further example of shearband formation are presented where the constitutive law is a classical viscoplastic one.

The sigma model on complex projective superspaces

The sigma model on complex projective superspaces CP^{S-1|S} gives rise to a continuous family of interacting 2D conformal field theories which are parametrized by the curvature radius R and the theta angle \theta. Our main goal is to determine the spectrum of the model, non-perturbatively as a function of both parameters. We succeed to do so for all open boundary conditions preserving the full global symmetry of the model. In string theory parlor, these correspond to volume filling branes that are equipped with a monopole line bundle and connection. The paper consists of two parts. In the first part, we approach the problem within the continuum formulation. Combining combinatorial arguments with perturbative studies and some simple free field calculations, we determine a closed formula for the partition function of the theory. This is then tested numerically in the second part. There we propose a spin chain regularization of the CP^{S-1|S} model with open boundary conditions and use it to determine the spectrum at the conformal fixed point. The numerical results are in remarkable agreement with the continuum analysis.

Spatially Localized Synchronous Oscillations in Synaptically Coupled Neuronal Networks: Conductance-based Models and Discrete Maps

We study the qualitative behavior of localized synchronous oscillations organized by synaptic inhibition in two types of spatially extended neuronal network models driven by a time-independent, localized excitatory input. Each network is formulated as a one-dimensional network of conductance-based models constituting a high-dimensional dynamical system of nonlocal differential equations. Although such equations readily generate highly complex dynamic behavior, in the case of strong inhibitory coupling the response of the network to a localized Gaussian input is a solution in which a single, continuous band of cells fire nearly synchronous action potentials, in an approximately periodic fashion in time. Tracking the cycle-to-cycle evolution of the width of the band of synchronous action potentials reveals the characteristic behavior of low-dimensional, discrete dynamical systems. Based upon a continuum formulation of the conductance-based model, we heuristically develop and analyze one- and two-dimensional implicit discrete maps for both a purely inhibitory and an excitatory-inhibitory network of neurons. Although the discrete maps do not predict the band widths precisely, they generally reflect the qualitative behavior of the conductance-based model. The most salient features of the bifurcations of fixed points to period 2 orbits and resonances indicate that in some cases these high-dimensional continuous dynamical systems exhibit behavior which can be captured in related low-dimensional discrete maps. Finally, we describe a global bifurcation in the discrete map for the excitatory-inhibitory network in which a strong (1:2) resonance bifurcation occurs on a period 2 orbit, giving rise to a pair of double homoclinic tangles that generate nontrivial dynamics.

Nonlinear elastic behavior of graphene:Ab initiocalculations to continuum description

The nonlinear in-plane elastic properties of graphene are calculated using density-functional theory. A thermodynamically rigorous continuum description of the elastic response is formulated by expanding the elastic strain energy density in a Taylor series in strain truncated after the fifth-order term. Upon accounting for the symmetries of graphene, a total of fourteen nonzero independent elastic constants are determined by least-squares fit to the ab initio calculations. The nonlinear continuum description is valid for infinitesimal and finite strains under arbitrary in-plane tensile loading in circumstance for which the bending stiffness can be neglected. The continuum formulation is suitable for incorporation into the finite element method.

Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation

A thermal–mechanical multiresolution continuum theory is applied within a finite element framework to model the initiation and propagation of dynamic shear bands in a steel alloy. The shear instability and subsequent stress collapse, which are responsible for dynamic adiabatic shear band propagation, are captured by including the effects of shear driven microvoid damage in a single constitutive model. The shear band width during propagation is controlled via a combination of thermal conductance and an embedded evolving length scale parameter present in the multiresolution continuum formulation. In particular, as the material reaches a shear instability and begins to soften, the dominant length scale parameter (and hence shear band width) transitions from the alloy grain size to the spacing between micro-voids. Emphasis is placed on modeling stress collapse due to micro-void damage while simultaneously capturing the appropriate scale of inhomogeneous deformation. The goal is to assist in the microscale optimization of alloys which are susceptible to shear band failure.

Spontaneous dissipation of elastic energy by self-localizing thermal runaway

Thermal runaway instability induced by material softening due to shear heating represents a potential mechanism for mechanical failure of viscoelastic solids. In this work we present a model based on a continuum formulation of a viscoelastic material with Arrhenius dependence of viscosity on temperature and investigate the behavior of the thermal runaway phenomenon by analytical and numerical methods. Approximate analytical descriptions of the problem reveal that onset of thermal runaway instability is controlled by only two dimensionless combinations of physical parameters. Numerical simulations of the model independently verify these analytical results and allow a quantitative examination of the complete time evolutions of the shear stress and the spatial distributions of temperature and displacement during runaway instability. Thus we find that thermal runaway processes may well develop under nonadiabatic conditions. Moreover, nonadiabaticity of the unstable runaway mode leads to continuous and extreme localization of the strain and temperature profiles in space, demonstrating that the thermal runaway process can cause shear banding. Examples of time evolutions of the spatial distribution of the shear displacement between the interior of the shear band and the essentially nondeforming material outside are presented. Finally, a simple relation between evolution of shear stress, displacement, shear-band width, and temperature rise during runaway instability is given.

A Computational Design Methodology for Assembly and Actuation of Thin-Film Structures via Patterning of Eigenstrains

We develop a computational approach to design 3-D structures that can be fabricated and then assembled and/or actuated by spatially tailoring the layout of multilayer films with eigenstrains. Eigenstrains are stress-free strains when they occur in an unconstrained solid. They are almost an inevitable companion, albeit often unwanted, of thin-film processes. When they vary through the thickness, the constraint of the layers leads to internal stresses and bending and buckling deformations can occur; when they additionally vary in the plane of the film, more complex deformations can result. To advantageously use this phenomenon, we build on relatively simple mechanics ideas in a continuum formulation and combine geometrically nonlinear finite-element analysis of arbitrary-shaped multilayer films with a topology optimization methodology to determine the material layout in each layer so the film deforms into a prescribed shape. We expand our previous experimentally validated approach to include initially curved films and anisotropic eigenstrains. Using an extended system formulation for directly computing instability points allows us to tailor postbuckling response while explicitly controlling the design at limit and bifurcation points. We demonstrate the potential and versatility of our approach by applying it to a series of problems of contemporary and emerging interest.