Continuum Formulation Research Articles

Mirrored continuum and molecular scale simulations of the ignition of high-pressure phases of RDX.

We present a mirrored atomistic and continuum framework that is used to describe the ignition of energetic materials, and a high-pressure phase of RDX in particular. The continuum formulation uses meaningful averages of thermodynamic properties obtained from the atomistic simulation and a simplification of enormously complex reaction kinetics. In particular, components are identified based on molecular weight bin averages and our methodology assumes that both the averaged atomistic and continuum simulations are represented on the same time and length scales. The atomistic simulations of thermally initiated ignition of RDX are performed using reactive molecular dynamics (RMD). The continuum model is based on multi-component thermodynamics and uses a kinetics scheme that describes observed chemical changes of the averaged atomistic simulations. Thus the mirrored continuum simulations mimic the rapid change in pressure, temperature, and average molecular weight of species in the reactive mixture. This mirroring enables a new technique to simplify the chemistry obtained from reactive MD simulations while retaining the observed features and spatial and temporal scales from both the RMD and continuum model. The primary benefit of this approach is a potentially powerful, but familiar way to interpret the atomistic simulations and understand the chemical events and reaction rates. The approach is quite general and thus can provide a way to model chemistry based on atomistic simulations and extend the reach of those simulations.

A phase-field approach to model fracture of arterial walls: Theory and finite element analysis

This study uses a recently developed phase-field approach to model fracture of arterial walls with an emphasis on aortic tissues. We start by deriving the regularized crack surface to overcome complexities inherent in sharp crack discontinuities, thereby relaxing the acute crack surface topology into a diffusive one. In fact, the regularized crack surface possesses the property of Gamma-Convergence, i.e. the sharp crack topology is restored with a vanishing length-scale parameter. Next, we deal with the continuous formulation of the variational principle for the multi-field problem manifested through the deformation map and the crack phase-field at finite strains which leads to the Euler–Lagrange equations of the coupled problem. In particular, the coupled balance equations derived render the evolution of the crack phase-field and the balance of linear momentum. As an important aspect of the continuum formulation we consider an invariant-based anisotropic constitutive model which is additively decomposed into an isotropic part for the ground matrix and an exponential anisotropic part for the two families of collagen fibers embedded in the ground matrix. In addition we propose a novel energy-based anisotropic failure criterion which regulates the evolution of the crack phase-field. The coupled problem is solved using a one-pass operator-splitting algorithm composed of a mechanical predictor step (solved for the frozen crack phase-field parameter) and a crack evolution step (solved for the frozen deformation map); a history field governed by the failure criterion is successively updated. Subsequently, a conventional Galerkin procedure leads to the weak forms of the governing differential equations for the physical problem. Accordingly, we provide the discrete residual vectors and a corresponding linearization yields the element matrices for the two sub-problems. Finally, we demonstrate the numerical performance of the crack phase-field model by simulating uniaxial extension and simple shear fracture tests performed on specimens obtained from a human aneurysmatic thoracic aorta. Model parameters are obtained by fitting the set of novel experimental data to the predicted model response; the finite element results agree favorably with the experimental findings.

Multiscale modeling and simulation of brain blood flow.

The aim of this work is to present an overview of recent advances in multi-scale modeling of brain blood flow. In particular, we present some approaches that enable the in silico study of multi-scale and multi-physics phenomena in the cerebral vasculature. We discuss the formulation of continuum and atomistic modeling approaches, present a consistent framework for their concurrent coupling, and list some of the challenges that one needs to overcome in achieving a seamless and scalable integration of heterogeneous numerical solvers. The effectiveness of the proposed framework is demonstrated in a realistic case involving modeling the thrombus formation process taking place on the wall of a patient-specific cerebral aneurysm. This highlights the ability of multi-scale algorithms to resolve important biophysical processes that span several spatial and temporal scales, potentially yielding new insight into the key aspects of brain blood flow in health and disease. Finally, we discuss open questions in multi-scale modeling and emerging topics of future research.

Derivation of F=FeFp as the continuum limit of crystalline slip

In this paper we provide a proof of the multiplicative kinematic description of crystal elastoplasticity in the setting of large deformations, i.e. F=FeFp, for a two dimensional single crystal. The proof starts by considering a general configuration at the mesoscopic scale, where the dislocations are discrete line defects (points in the two-dimensional description used here) and the displacement field can be considered continuous everywhere in the domain except at the slip surfaces, over which there is a displacement jump. At such scale, as previously shown by two of the authors, there exists unique physically based definitions of the total deformation tensor F and the elastic and plastic tensors Fe and Fp that do not require the consideration of any non-realizable intermediate configuration and do not assume any a priori relation between them of the form F=FeFp. This mesoscopic description is then passed to the continuum limit via homogenization, i.e. by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. We show for two-dimensional deformations of initially perfect single crystals that the classical continuum formulation is recovered in the limit with F=FeFp, detFp=1 and G=CurlFp the dislocation density tensor.

Kinetic Models and Gas-Kinetic Schemes for Hybrid Simulation of Partially Rarefied Flows

Approaches to predict flowfields that display rarefaction effects incur a cost in computational time and memory that is considerably higher than methods commonly employed for continuum flows. For this reason, to simulate flowfields where continuum and rarefied regimes coexist, hybrid techniques have been introduced. In the present work, analytically defined gas-kinetic schemes based on the Shakhov and Rykov models, for monatomic and diatomic gas flows, respectively, are proposed and evaluated, with the aim of being used in the context of hybrid simulations. This should reduce the region where more expensive methods are needed by extending the validity of the continuum formulation. Moreover, because, for rarefied gas flows at high velocities, it is necessary to take into account the nonequilibrium among the internal degrees of freedom, the extension of the approach to employ diatomic gas models with rotational relaxation is a mandatory first step toward realistic simulations. Compared to the previous works the presented scheme is defined directly on the basis of kinetic models that involve a Prandtl number correction. Moreover, the methods are defined fully analytically instead of making use of Taylor expansion for the evaluation of the required derivatives. The scheme has been tested for various test cases and Mach numbers proving to produce reliable predictions in agreement with other approaches for near-continuum flows. Finally, the performance of the scheme, in terms of memory and computational time, compared to discrete velocity methods makes it a compelling alternative in place of more complex methods for hybrid simulations of weakly rarefied flows.

Holistic modeling of composites manufacturing using poromechanics

In the present paper, we present a novel finite-element method capable of handling most of the physics arising in the resin wet-out step for any composite system and processing case. The method is based on a compressible two-phase continuum formulation where a key feature is to model the involved physics via innovative use of the compressibility of the phases. On the one hand, the fluid-phase compressibility is used to capture the physics of the advancing resin front as well as the physics behind the flow front. On the other hand, solid-phase compressibility is used to model micro-infiltration of the resin and the corresponding preform compaction, essentially considered as a fluid sink problem. Finally, the generic porous media model is formulated in the finite strain regime. The model is implemented and demonstrated for different manufacturing methods and the results with respect to each example are presented. The degree of saturation, pressure distribution, preform deformation, and reaction forces are some of the post-processed results for different manufacturing methods. The ultimate goal of this contribution is to establish a unified generic and general simulation tool for structural (long fiber) composite processing where, to this date, there is no single FE-based tool available commercially for this purpose.

Sequential Homogenization of Reactive Transport in Polydisperse Porous Media

Direct numerical simulations of flow and transport in porous media are computationally prohibitive due to the disparity between the typical scale at which processes are well understood (e.g., the pore-scale) and the scale of interest (the system- or field-scale). Homogenization approaches overcome some of the difficulties of full pore-scale simulations by providing an upscaled representation of fine-scale processes. Real porous systems, e.g., rocks, pose additional challenges since they usually exhibit multimodal distributions in physical and chemical properties. Perforated domains, i.e., domains with impermeable inclusions embedded in a porous matrix, represent one such example. These hierarchical media cannot be approached by a single continuum formulation. Sequential homogenization techniques build a hierarchy of effective equations that sequentially carry the smallest scale information through the intermediate scales up to the macroscale. The advantage of sequential upscaling in handling multimodal distribution in physical and chemical properties lies in its computational efficiency compared to one-step homogenization: the information about smaller-scale heterogeneity is incorporated in the subsequent scales in terms of effective media properties. Yet, existence of one or multiple intermediate scales can significantly decrease the accuracy of multiscale formulations. We show that the accuracy of multiscale methods based on sequential upscaling is strongly influenced by a combination of geometric and dynamical scale separation conditions. In particular, we investigate under which conditions sequential homogenization of reactive solute transport in geometrically and chemically heterogeneous porous domains composed of bidisperse cylinders can accurately describe pore-scale processes. We show that under appropriate conditions, expressed in terms of Peźclet and Damkoźhler numbers and a scales separation parameter, the sequential upscaling method has second-order accuracy. We compare sequential upscaling results with the direct solution of the fully resolved pore-scale problem.

Thermodynamically consistent elasto-plastic microplane formulation for fiber reinforced concrete

In this work a thermodynamically consistent elasto-plastic microplane constitutive theory, aimed at simulating the failure behavior of Steel Fiber Reinforced Concrete (SFRC), is developed. The continuum (smeared crack) formulation, based on the microplane theory, assumes a parabolic maximum strength criterion in terms of normal and shear (micro-)stresses evaluated on each microplane to simulate the failure behavior of concrete. In the high confinement regime, a non-associated plastic flow rule is also defined in terms of microplane stresses. The well-known “Mixture Theory” is considered to account for the presence of fibers in concrete matrix. The interaction between steel fibers and cracked concrete in the form of fiber-to-concrete bond-slip and dowel mechanisms is taken into account. The complete formulation is fully consistent with the thermodynamic laws. After describing the proposed constitutive theory, numerical analyses at constitutive level of SFRC failure behavior are presented and discussed. Thereby, the variations of the fracture energy, post-peak strength and cracking behavior with the fiber contents are evaluated and compared against experimental data. The attention also focuses on the evaluation of the sensitivity of SFRC failure predictions with the proposed constitutive model regarding fiber orientation on one hand, and the bond-slip bridging actions and dowel mechanism on the other hand.

Biological tissue mechanics with fibres modelled as one-dimensional Cosserat continua. Applications to cardiac tissue

• Cosserat-fibre continuum-based transversely isotropic material law for biological soft tissue. • Investigation of the impact of higher-order stress and strain contributions in modelling of heart muscle tissue. • Demonstration of the improved robustness of the Cosserat-fibre approach as compared with a classical continuum formulation when applied to highly heterogeneous and non-uniform material anisotropy distributions. • Computational modelling of the rat left ventricle during passive filling. Classically, the elastic behaviour of cardiac tissue mechanics is modelled using anisotropic strain energy functions capturing the averaged behaviour of its fibrous micro-structure. The strain energy function can be derived via representation theorems for anisotropic functions where a suitable non-linear strain tensor, e.g. the Green strain tensor, describes locally the current state of strain ( Holzapfel, Ogden, 2009 , Zheng, 1994 ). These approaches are usually of phenomenological nature and do not elucidate on the complex heterogeneous material composition of cardiac tissue (Rosenberg and Cimrman, 2003). In this research the fibrous characteristics of the myocardium are modelled by one-dimensional Cosserat continua. This additionally allows for the inclusion of non-local effects due to the heterogeneous material composition at smaller scales. Specifically, the non-local material response is linked to higher-order deformation modes associated with twisting and bending of an assembly of muscle fibres arising from hierarchical multi-scale features within a representative volume element (RVE). In this sense, a scaling parameter characteristic for the tissue’s underlying micro-structure, becomes a material parameter of the formulation. As the anisotropic material composition of the myocardium throughout the heart is highly non-uniform, the ability to implicitly account for scale-dependent torsion and bending effects in the constitutive law gives this approach an advanced material description over classical formulations. The assumed hyperelastic material behaviour of myocardial tissue is represented by a non-linear strain energy function which includes contributions linked to the Cosserat -fibre continuum and complementary terms which refer to the extra-cellular matrix. Utilising the Element-free Galerkin method, simulations of specimen shear cubes and the left ventricle undergoing passive filling are introduced to investigate ventricular tissue mechanics.

Influence of Gribov ambiguities in a class of nonlinear covariant gauges

We consider Yang-Mills theories in a recently proposed family of nonlinear covariant gauges that consistently deals with the issue of Gribov ambiguities. Such gauges provide a generalization of the Curci-Ferrari-Delbourgo-Jarvis gauges which can be formulated as an extremization procedure and might be implemented in numerical calculations. This would allow for nonperturbative studies of Yang-Mills correlators in a broad class of covariant gauges continuously connected to the well-studied Landau gauge. We compute the ghost and gluon propagators in the continuum formulation at one-loop order in perturbation theory and we study their momentum dependence down to the deep infrared regime, with and without renormalization group improvement. In particular, we show that the theory admits infrared-safe renormalization-group trajectories with no Landau pole. Both the gluon and the ghost behave as massive fields at low energy, and the gluon propagator is transverse even away from the Landau gauge limit. We compare our results to those obtained in the usual Curci-Ferrari model, which allows us to pinpoint the specific effects arising from our treatment of Gribov copies.

Substrate Melting and Re-solidification During Impact of High-Melting Point Droplet Material

In this paper, impact, spreading, and solidification of molten droplet on a dissimilar substrate along with substrate melting and its re-solidification are investigated numerically. Volume of fluid surface tracking method coupled with the solidification model within a one-domain continuum formulation is used to model the transient flow during the droplet impact, its subsequent spreading, and solidification. Evolution of melting front in the substrate is modeled by solving the governing equations for solidification in the substrate, too. Simulations are performed for the impact of a heated droplet on a substrate. The model predicts substrate melting, which can give better insight of bonding between the coating material and substrate, where droplet and substrate are of different materials. It is observed that melting in the substrate in the present case starts soon after the impact of the heated droplet. The depth and the width of the melting front in the substrate increase with the time and after reaching a maximum they start to decrease because of start of re-solidification from the melted edge. In the central part of the splat droplet solidifies, while the substrate remains melted which can enhance the coating strength and its bonding with the substrate.

Bifurcation investigations of coupled damage-plasticity models for concrete materials

Abstract This communication addresses the localization properties of a coupled damage-plasticity formulation for concrete materials to provide information on the onset of material bifurcation and the critical failure modes. Two separate loading functions are considered, one for damage and one for plasticity. A three-invariant yield surface is used to model plasticity and to consider the significant role of the intermediate principal stress and the Lode parameter on the failure of concrete materials. A non-associated flow rule is employed to control inelastic dilatancy. To model degradation of the elastic stiffness a scalar-valued isotropic damage formulation is introduced based on the total strain energy formulation that is used. Monotonic and cyclic uniaxial compression experiments are performed on concrete cylinders under displacement control and photogrammetric images are collected for Digital Image Correlation Analysis. The triaxial based damage-plasticity model is calibrated based on these experimental observations and is implemented in Matlab. Extensive localization analysis studies are performed at the constitutive level for representative load scenarios in the form of non-positive properties of the elastoplastic-damage localization tensor. The contributions of damage, plasticity and coupled damage-plasticity are explored and compared for classical Boltzmann and Micropolar Cosserat continuum formulations.

Topology design for maximization of fundamental frequency of couple-stress continuum

In this paper, the topology optimization formulation of couple-stress continuum is investigated for maximizing the fundamental frequency. A modified bound formulation is used to prevent the order s...

Nineteenth century molecular models with a glance at modern discrete—continuum theories

AbstractThe origins of multiscale modelling, lying in the classical molecular models of Nineteenth century originally formulated to provide explanations per causas of elasticity, are investigated with the aim of drawing suggestions for the modelling of new complex materials. Conceptual guidelines for deriving scale dependent continuum formulations, here continuum models with additional degrees of freedom (multifield continua), are defined starting from non‐classical corpuscular descriptions and the historical and epistemological settings of the current modelling proposals are highlighted. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A novel strain energy relationship for red blood cell membrane skeleton based on spectrin stiffness and its application to micropipette deformation.

Red blood cell (RBC) membrane skeleton is a closed two-dimensional elastic network of spectrin tetramers with nodes formed by short actin filaments. Its three-dimensional shape conforms to the shape of the bilayer, to which it is connected through vertical linkages to integral membrane proteins. Numerous methods have been devised over the years to predict the response of the RBC membrane to applied forces and determine the corresponding increase in the skeleton elastic energy arising either directly from continuum descriptions of its deformation, or seeking to relate the macroscopic behavior of the membrane to its molecular constituents. In the current work, we present a novel continuum formulation rooted in the molecular structure of the membrane and apply it to analyze model deformations similar to those that occur during aspiration of RBCs into micropipettes. The microscopic elastic properties of the skeleton are derived by treating spectrin tetramers as simple linear springs. For a given local deformation of the skeleton, we determine the average bond energy and define the corresponding strain energy function and stress-strain relationships. The lateral redistribution of the skeleton is determined variationally to correspond to the minimum of its total energy. The predicted dependence of the length of the aspirated tongue on the aspiration pressure is shown to describe the experimentally observed system behavior in a quantitative manner by taking into account in addition to the skeleton energy an energy of attraction between RBC membrane and the micropipette surface.

First-order aggregation models and zero inertia limits

We consider a first-order aggregation model in both discrete and continuum formulations and show rigorously how it can be obtained as zero inertia limits of second-order models. In the continuum case the procedure consists in a macroscopic limit, enabling the passage from a kinetic model for aggregation to an evolution equation for the macroscopic density. We work within the general space of measure solutions and use mass transportation ideas and the characteristic method as essential tools in the analysis.

Dynamic shear force amplification in regular frame–wall systems

SummaryA parametric study is conducted to investigate the dynamic shear amplification factor (DAF) in low‐to‐mid‐rise frame–wall systems in which the reinforcement curtailment along the height matches the required code strength. The level of frame–wall interaction is varied by changing the wall index, defined as the ratio of the total wall area to the floor plan area, in a generic frame–wall system, and its correlation with the DAF is investigated. Wall index values ranging in the 0.2% to 2% interval are selected. Walls with lengths of 3m , 5 m and 8 m are used in the design of model buildings of 4, 8 and 12 stories. Shear–flexure beam continuum formulation is used in design and modeling. The global behavior is analyzed using nonlinear response history procedure using spectrum compatible ground motions. It is found that the primary source of amplification is the level of inelastic demand on the system. Walls designed for code‐specified force reduction factor R = 6 experienced an average base shear force amplification in the order of 1.64 with standard deviation of 0.19 with respect to design shear force. Amplification diminishes with decreasing R. An expression for the dynamic amplification factor as a function of the number of stories and force reduction factor R is proposed. Copyright © 2015 John Wiley & Sons, Ltd.

Energy dissipation in the course of the fatigue degradation: Mathematical derivation and experimental quantification

This paper is concerned with a continuum formulation for irreversible energy dissipation that accounts for generated acoustic emissions during the loading of the materials. Within a thermodynamically consistent framework, the coupling between the mechanical, thermal and acoustic fields is formulated. The evolution of the dissipative energy is experimentally measured as the material is degraded. A statistically similar behavior is observed in different forms of the dissipated energy as the material degrades.

Effect of the Gribov horizon on the Polyakov loop and vice versa

We consider finite-temperature SU(2) gauge theory in the continuum formulation, which necessitates the choice of a gauge fixing. Choosing the Landau gauge, the existing gauge copies are taken into account by means of the Gribov–Zwanziger quantization scheme, which entails the introduction of a dynamical mass scale (Gribov mass) directly influencing the Green functions of the theory. Here, we determine simultaneously the Polyakov loop (vacuum expectation value) and Gribov mass in terms of temperature, by minimizing the vacuum energy w.r.t. the Polyakov-loop parameter and solving the Gribov gap equation. Inspired by the Casimir energy-style of computation, we illustrate the usage of Zeta function regularization in finite-temperature calculations. Our main result is that the Gribov mass directly feels the deconfinement transition, visible from a cusp occurring at the same temperature where the Polyakov loop becomes nonzero. In this exploratory work we mainly restrict ourselves to the original Gribov–Zwanziger quantization procedure in order to illustrate the approach and the potential direct link between the vacuum structure of the theory (dynamical mass scales) and (de)confinement. We also present a first look at the critical temperature obtained from the refined Gribov–Zwanziger approach. Finally, a particular problem for the pressure at low temperatures is reported.

Regularization methods for Nuclear Lattice Effective Field Theory

We investigate Nuclear Lattice Effective Field Theory for the two-body system for several lattice spacings at lowest order in the pionless as well as in the pionful theory. We discuss issues of regularizations and predictions for the effective range expansion. In the pionless case, a simple Gaussian smearing allows to demonstrate lattice spacing independence over a wide range of lattice spacings. We show that regularization methods known from the continuum formulation are necessary as well as feasible for the pionful approach.