Additional Balance Equation Research Articles

A continuous–discontinuous localizing gradient damage framework for failure analysis of quasi-brittle materials

This article presents a continuous–discontinuous approach that provides a discontinuous character to the quasi-brittle fracture process, modeled in a continuous setting through a micromorphic stress-based localizing gradient damage framework. The discontinuous framework is developed by enhancing the problem fields using discontinuous interpolations in the critically damaged regions exploiting local partition of unity via the eXtended Finite Element Method (XFEM). The proposed approach incorporates an improved spatial nonlocal diffusive behavior in the continuous bulk through an additional microforce balance equation to ensure an improved prediction of the initial phase of the loading process. Level sets are computed to track the discontinuity within the finite element mesh, predicted via evolving micromorphic equivalent strain. Here, the proposed continuous–discontinuous approach combines the advantages of the gradient damage method and the XFEM while exhibiting improved convergence using low-order finite elements during numerical simulations. The enhanced numerical kinematics of the proposed approach successfully eliminates damage spreading during the final failure stages and provides an accurate description of a true discontinuity without using cohesive zone modeling. Various representative structural examples of benchmark tests involving varying loading conditions are investigated to demonstrate the accuracy and robustness of the proposed approach to provide qualitative and quantitative accurate numerical predictions.

On the implementation of rate‐independent gradient‐enhanced crystal plasticity theory

AbstractIn the present work, classical local crystal plasticity is reformulated using incremental energy minimization. The resulting constrained optimization problem is efficiently solved by a nonlinear complementary problem, which additionally solves the problem of determination of the set of active slip systems. The model is extended to a nonlocal, gradient‐enhanced, formulation by employing and adapting the micromorphic approach. By doing so, the implementation of the gradient‐enhanced model is almost identical to that of the original local one, with the exception of additional balance equations.

Prediction of Large Format High Power Cell Performance Using NTG Model

Accurate prediction of cell performance using electrochemical parameters has always been a challenge in terms of computational complexity and collection of parameters for individual cell components (such as cathode, anode and electrolyte). In construction of a completely physics based model, the parameterisation is tedious and extremely time consuming. In order to save time and maintain accuracy, an alternate mathematical model which demands less experimental inputs especially from full cell alone is of interest in this study. In addition, the model which offers scope to manipulate the geometrical features such as tab positions and cell dimensions to further optimise the cell performance at minimal computational time is desirable. NTG model is a semi empirical model which predicts the transient behaviour of lithium ion batteries. In this study, two significant full cell parameters namely effective conductance and open circuit voltage are determined for a high power cell by stepped pulse experiments over a wide range of operating temperatures between -25°C and 50°C. Other parameters such as cell thermal conductivity, specific heat capacity and density are also determined. Additional heat balance equations are incorporated to simulate the thermal behaviour of cell. The model is constructed using partial differentiation equations in COMSOL and solved across a representative 3D geometry. Interestingly, the electrochemical performance curves and the cell surface temperature from simulations are quite close to that of experiments. The feasibility of potential, current density and temperature distribution in three dimensions in this model give better insights into the underlying battery physics. Although the diffusion of lithium ions is overlooked in this model for simplicity, the model offers the advantage of faster simulations for high power cells where the electrode loading is lesser.

Localizing gradient damage model with smoothed stress based anisotropic nonlocal interactions

In this paper, a localizing gradient damage model with smoothed stress based anisotropic nonlocal interactions is proposed, which overcomes the limitations of conventional damage models for the accurate prediction of localized failure of quasi-brittle materials. The proposed model uses a modified anisotropic nonlocal interaction domain, which also accounts the decrease in nonlocal interactions during the softening regime. The modification in the nonlocal interaction domain is done by introducing a normalized anisotropic interaction kernel and a damage dependent interaction function in the constitutive framework. The normalized anisotropic interaction kernel governs the orientation of the nonlocal interaction domain, depending on the principal stress state, whereas the interaction function controls the intensity of the nonlocal interactions. Due to these enhancements in the interaction behavior, an anisotropic behavior is introduced in the nonlocal interactions. The proposed model employs a thermomechanical micromorphic procedure, where an additional microforce balance equation is introduced in the constitutive framework apart from the standard equilibrium equation. To enable the use of low order finite elements, an integral nonlocal smoothening technique is employed to obtain a smoothed stress field for the better prediction of diffusive interaction domain. For the numerical implementation purpose, the performance and the accuracy of the proposed model are tested against mode-I, mode-II and mixed mode type of failure problems under plane deformation condition.

A robust incompressible Navier-Stokes solver for high density ratio multiphase flows

This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined staggered Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The algorithm combines the interface capturing level set method with a variable-coefficient incompressible Navier-Stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude. The discretization approach ensures second-order pointwise accuracy for both velocity and pressure with several physical boundary treatments, including velocity and traction boundary conditions. The paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver. To ensure the stability of the numerical scheme in the presence of high density and viscosity ratios, we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations. This consistency is achieved by solving an additional mass balance equation, which we approximate via a strong stability preserving Runga-Kutta time integrator and by employing the same mass flux (obtained from the mass equation) in the discrete momentum equation. The scheme uses higher-order total variation diminishing (TVD) and convection-boundedness criterion (CBC) satisfying limiter to avoid numerical fluctuations in the transported density field. The high-order bounded convective transport is done on a dimension-by-dimension basis, which makes the scheme simple to implement. We also demonstrate through several test cases that the lack of consistent mass and momentum transport in non-conservative formulations, which are commonly used in practice, or the use of non-CBC satisfying limiters can yield very large numerical error and very poor accuracy for convection-dominant high density ratio flows. Our numerical scheme also uses well-balanced surface tension and gravity force discretizations. In the hydrostatic limit, we show that the well-balanced formulation mitigates spurious flow currents and achieves discrete force-balance between the pressure gradient and surface tension or gravity.

On the implementation of finite deformation gradient-enhanced damage models

We introduce a comprehensive framework for the efficient implementation of finite deformation gradient-regularised damage formulations in existing finite element codes. The numerical implementation is established within a thermo-mechanically fully coupled finite element formulation, where the heat equation solution capabilities are utilised for the damage regularisation. The variationally consistent, gradient-extended and geometrically non-linear damage formulation is based on an overall free energy function, where the standard local free energy contribution is additively extended by two non-local terms. The first additional term basically contains the referential gradient of the non-local damage variable. Secondly, a penalty term is added to couple the local damage variable—the evolution of which is governed by an ordinary differential equation—and the non-local damage field variable that is governed by an additional balance equation of elliptic type.

A novel simplified multivariate PBE solution method for mass transfer problems

Interphase mass transfer estimation may require not only the accurate knowledge of the interfacial area, which depends on the information about the size of each dispersed element, but also on the driving force, that can be different if the elements of the disperse phase have different chemical composition. To take into account such polydispersity, bivariate (or multivariate) population balance model (PBM) are formulated according to physical phenomena occurring in the investigated mass transfer problem. This often includes aggregation, breakage, advection, mass transfer of the chemical species and chemical reactions of the transferring components. In this work we propose a novel and simplified method to solve the bivariate/multivariate population balance equation for a mass transfer problem, based on the high-order moment-conserving method of classes (HMMC) (Alopaeus et al., 2006). The proposed method is based on the idea of deriving additional material balance equations for the concentration of the droplets belonging to each size class, reducing significantly the total number of unknown variables with respect to true bivariate/multivariate method of classes. This modeling approach is compared with two other possible solution methods for a test case in which mass transfer and chemical reactions occur in a system with two immiscible liquid phases. In the first the traditional approach is used, where a single material balance is formulated for the disperse phase along with PBM, while in the second a true bivariate/multivariate solution method is used. The results of this comparison show that the proposed method is robust and accurate, capable of properly describing the multidimensional droplet size-composition distribution needed to evaluate the mass transfer rates, in a fraction of the computational time compared with more accurate methods.

Finite element analysis of internally balanced elastic materials

This paper provides a finite element treatment for a generalized hyperelastic theory in which the deformation gradient is subject to the multiplicative two-factor decomposition that is commonly used in large strain plasticity. The multiplicative decomposition in the present context is motivated by a theory of balanced internal variables, one of which is the first factor in the decomposition. Previous studies have shown how the associated continuum mechanical formulation then leads not only to the stress equations of equilibrium but also to an additional tensor balance equation that serves to determine the individual factors in the decomposition. Such a theory has been shown to predict loading thresholds associated with the emergence of slip surfaces at grip locations in well-posed boundary value problems. However, previous numerical treatments focused on shooting method procedures that failed to resolve loading plateaus and other key features associated with the emergence of such singular surfaces. In the present work we describe a finite element treatment that resolves such previously obscure details. Additional finite element simulations illustrate new aspects of singular surface emergence in this type of theory, including the emergence of singular surfaces in the interior of the computational domain.

A New Nodal Expansion Method with High-Order Moments for the Reduction of Numerical Oscillation in Convection-Diffusion Problems

The nodal integral method (NIM) has been widely used to solve multidimensional steady-state convection-diffusion problems. However, unphysical oscillating behavior arises when NIM is applied to steep-gradient problems and discontinuous problems. In this paper, a new nodal expansion method (NEM) with high-order moments (NEM_HM) is developed to reduce the numerical oscillation drawback of NIM. High-order moments of transverse-integrated variables are introduced. Based on the definition of Legendre moments, all the expansion coefficients of NEM_HM can be defined as shared moments and unshared moments. Then, the calculation framework of the traditional NEM is extended to include the high-order moments. Additional nodal balance equations are introduced to ensure the uniqueness of all the shared variables such as node-average variables. Finally, coupled discrete equations are obtained in terms of various order moments on the surfaces of the nodes. The classical Smith-Hutton problem and a cross-flow problem are chosen to test the effectiveness of NEM_HM. Numerical results show that the accuracy of NEM_HM outperforms NIM for steep-gradient problems and discontinuous cases.

Nonlocal constitutive equations of elasto-visco-plasticity coupled with damage and temperature

In this paper, the nonlocal anisothermal elasto-visco-plastic constitutive equations strongly coupled with ductile isotropic damage, nonlinear isotropic hardening and kinematic hardening are developed to model the material behaviour under finite strain. The new micromorphic variable of damage is introduced into the principle of virtual power and new additional balance equations are obtained. Thermodynamically-consistent nonlocal constitutive equations are then deduced. The evolution equations are deduced from the generalized normality rule for the Norton-Hoff visco-plastic potential. This model is used to simulate various material responses under different velocities at high temperature. The micromorphic parameters of damage: micromorphic density and H moduli are studied to examine the effects of micromorphic damage. Biaxial tension is performed to make a comparison between the local damage model and the micromorphic damage model.

A mixture model of poroelastic materials

Abstract Chemically-reacting porous materials are investigated within the theory of mixtures. The key idea is that for incompressible constituents, the volume fractions satisfy balance equations strictly analogous to those of mass densities. As a consequence, upon compatibility with thermodynamics, constitutive models are established where the evolution of the volume fractions is governed by the conservation of mass and need not be given by additional balance equations. Next an analogous scheme is developed for materials with voids where chemical reactions no longer occur. Compressible constituents are also considered and the concentrations are used as the variables characterizing the immiscibility. Some comments are given about the approaches where further fields occur, such as equilibrated forces and stresses, and are associated with additional balance equations.

Modelling Inhomogeneous Mechanical Properties in Adhesive Bonds

Polymers often show an inhomogeneous distribution of their mechanical properties. In adhesive layers these inhomogeneities result from the formation of interphases close to the substrates. In addition, chemical aging processes are driven by diffusive processes entering the specimen from the boundaries and changing the local properties of the polymer. In the present contribution a model is presented that takes into account the local effects of the interphase's formation or of the aging processes, respectively. The model is based on a phenomenological continuum mechanical approach, i.e., the effects are described by the introduction of an additional field in combination with an additional balance equation and corresponding boundary conditions. In the first part of this paper the formation of interphases is investigated. Comparing the model with experimental results allows for the identification of the model parameters. In the second part the model is enhanced to describe effects due to chemical aging. Some illustrative examples show the general behaviour of the model and motivate a series of experiments allowing for the required parameter identification.

Large eddy simulation of a growing turbulent premixed flame kernel using a dynamic flame surface density model

The relevant description of a flame kernel growth in homogeneous isotropic turbulence environment is a key ingredient for the simulation of combustion in spark-ignited internal combustion engine. A small flame kernel burns at an increasing rate as it grows and becomes progressively more and more wrinkled by the prevailing turbulence. Large eddy simulation (LES) models based on algebraic expressions for the turbulent flame speed, the flame surface density or the flame wrinkling factor cannot reproduce such a situation because they assume an equilibrium between turbulence and flame wrinkling which is generally not reached during the simulation as the flame is laminar at early stages. A possible solution is to solve an additional balance equation for the flame surface density or for the wrinkling factor but a promising alternative is to implement a dynamic model where the wrinkling factor, measuring the subgrid scale flame surface, is automatically determined along the computation from the known instantaneous resolved flame front. A Germano-like identity expresses the conservation of the total flame surface at filter and test filter scales to automatically adjust the parameter of a fractal-like wrinkling factor model from the instantaneous resolved flame front characteristics. This model is tested against the experimental data of Renou et al. (2000) [33] to reproduce the temporal evolution of the flame kernel growth. For a range of ignition parameters, preliminary results are in good agreements with experimental data and show that the model parameter evolves with time and depends on operating conditions, validating the dynamic procedure.

A multi-field incremental variational framework for gradient-extended standard dissipative solids

The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time–space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.

Extending the inelastic thermal spike model for semiconductors and insulators

The inelastic thermal spike framework was extended to incorporate an additional balance equation for the carrier density. Temporal and spatial evolution of carrier density, electronic and lattice temperatures were solved for silicon using a finite difference method. Calculated track radii for a range of electronic stopping powers are presented. The model allows us to fit the electron–phonon coupling to experimental data of amorphised track radii. We compare the methodology of this framework to an earlier inelastic thermal spike model, which is based on the two-temperature model for non-equilibrium processes in metals, and discuss its contribution to the understanding of microscopic processes following a swift ion irradiation event in band gap materials.

Size Effects due to the Formation of Interphases in Polymer Joints

AbstractIn this contribution we investigate the numerical modelling of polymer bonds under quasi‐static and isothermal conditions. From experimental investigations it is known that polymers form interphases in contact with substrates. These interphases influence the macroscopic material behaviour in the form of size effects, which are observed either in the form “Smaller is Stiffer” or “Smaller is Weaker” depending on the substrate. Taking the formation of the microstructure into account, we postulate an additional balance equation based on an abstract scalar‐valued structure parameter which offers us a way to consider the interphases. We investigate the coupled model consisting of the balance of linear momentum and the balance of the structure parameter by finite element analysis. Different numerical examples are shown. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Entropy production calculation for turbulent shear flows and their implementation in cfd codes

Entropy production in turbulent shear flows with heat transfer is calculated locally and afterwards integrated over the whole flow domain. This quantity can serve as a parameter to determine the efficiency of turbulent heat transfer processes. Based on the time averaged entropy balance equation, four different mechanisms of entropy production can be identified and cast into mathematical equations. They are: dissipation in the mean and the fluctuating velocity fields and heat flux due to the mean and the fluctuating temperature fields. It turns out that no additional balance equation has to be solved, provided the turbulent dissipation rate is known in the flow field together with the mean velocity and temperature distribution. Since all four entropy production rates show very steep gradients close to the wall numerical solutions are far more effective with wall functions for the production terms. These wall functions are mandatory when high Reynolds number turbulent models are used, as for example the high Reynolds number k– ε model, like in our case. As an example, flow through a heated pipe with a twisted tape inserted is calculated in detail including the local entropy production rate. For this configuration experimental results show an increase in heat transfer as well as in pressure drop when the spiral slope of the twisted tape is increased. Therefore, no optimum of the spiral slope can be found in the experiments. An analysis based on entropy production, however, reveals that there is a distinct optimum for a certain slope of the twisted tape. Thus, entropy production can be used as an efficiency parameter with respect to minimizing the loss of available work in a process.

Nonlinear projection methods to approximate combustion turbulent flows

The paper deals with the numerical resolution of the convection–diffusion–reaction system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDEs, that governs such a model, turns out to be in non-conservative form and usual numerical approaches grossly fail in the capture of viscous shock layers. We propose a new method, the so-called nonlinear procedure, for correcting the errors while preserving all the stability properties.

An approximate nonlinear projection scheme for a combustion model

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite volume methods induce large errors when approximated the convection-diffusion extracted system. To solve this difficulty, recent works propose a nonlinear projection scheme based on cancellation phenomenon of relevant dissipation rates of entropy. Unfortunately, such a property never holds in the present framework. The nonlinear projection procedures are thus extended.

Strain Gradient Crystal Plasticity: Thermomechanical Formulations and Applications

Since the pioneering work of [3] (see [4] for a review), strain gradient plasticity has aroused increasing interest in the mechanics of materials community, leading to a large panel of non local plasticity models. However, comparisons between these models remain seldom [19], and he full thermomechanical framework is usually not provided. The attention is focused here on models incorporating modified or additional balance equations and therefore additional boundary conditions in order to solve practical boundary value problems. The reader is referred to [1] for models sticking to classical structure of the boundary value problem. The aim of the present work is to bring together and to some extent reconcile three main trends of strain gradient plasticity models applied to metal single crystals : the gradient of internal variable, second grade and Cosserat approaches. In particular, the thermodynamical setting for all three models will be described in a unified way, in order to make actual differences between theories really visible. For simplicity and clarity, the analysis is restricted to single slip and rate–independent constitutive equations. Extensions to the general viscoplastic multislip case are straightforward, based on multicriterion elastoviscoplasticity. Each model presentation follows the three following main steps :