Computational homogenization of liquid-phase sintering with seamless transition from macroscopic compressibility to incompressibility

Liquid phase sintering of particle agglomerates is modeled on the mesoscale as the viscous deformation of particle-particle contact, whereby the single driving force is the surface tension on the particle/pore interface. On the macroscale, a quasistatic equilibrium problem allows for the prediction of the shrinkage of the sintering body. The present paper presents a novel FE2 formulation of the two-scale sintering problem allowing for the transition to zero porosity, implying macroscale incompressibility. The seamless transition from compressibility to incompressibility on the macroscale is accomplished by introducing a mixed variational format. This has consequences also for the formulation of the mesoscale problem, that is complemented with an extra constraint equation regarding the prolongation of the volumetric part of the macroscopic rate-of-deformation. The numerical examples shows the sintering of a single representative volume element (RVE) which is sheared beyond the point where the porosity vanishes while subjected to zero macroscopic pressure. © 2013 The Authors

Atomistic-to-continuum coupling based on goal-oriented adaptivity and quasi-continuum approximation

It has long been understood that some of the key characteristics of materials, like grain boundaries, dislocation cores, and crack tips cannot be modeled realistically within the framework of continuum mechanics. It is believed that defects and surface effects play important roles at small scales, where the continuum assumptions are violated. Therefore, further information on a material at its atomic scale is required to give a more physically based description of the phenomena governing the macroscopic behavior. A well-established method for concurrent coupling between atomistic and continuum regions is the quasi-continuum (QC) method. Basically, it relies on the fact that in many practical cases, only a small region in the material needs to be modeled atomistically. As a result, the assumptions of continuum mechanics can still be adopted for modeling the remaining regions without loss of the accuracy. This strategy provides a significant gain in computational cost, whereas the interesting phenomena pertaining to the atomistic area are also preserved. In this study, atomistic-to-continuum homogenization of molecular statics problem has been advanced with particular focus on the effect of lattice defects. A representative unit lattice (RUL) with Cauchy–Born boundaries is considered for obtaining homogenized response of the graphene lattice in terms of membrane forces (macroscale stress). In order to facilitate such an analysis, an adaptive strategy based on goal-oriented error estimation is developed for retaining the accuracy in the “goal-quantity” or “quantity of interest”, chosen to be the macroscale (continuum) stress. The QC method is introduced in two steps. First, we consider the restriction of atom displacement in terms of the representative atoms as a model reduction. The second step in the QC method is that of quadrature. For large QC elements, i.e., for a large amount of atoms whose placements are governed by the same representative atoms, the bond energy and its derivatives are typically computed using an appropriate discrete quadrature. We show how cluster approximation generates a quadrature error (in addition to the discretization error). Error because of different sources, namely: interpolation and quadrature parts of the QC method is approximated by solving the pertinent dual (adjoint) problem relevant to the output of interest. A hierarchical strategy is proposed for approximating the residual on an intermediate mesh, thus avoiding high computational cost that pertains to solving the full system. Homogenization of the macroscale membrane forces, including initial relaxation, is considered for the graphene in presence of divacancy defect. The Carbon–Carbon interactions are modeled via the Tersoff–Brenner potential enabling computation of bond energies up to the next nearest neighbor. The reliability of the adaptive algorithm is demonstrated through the numerical simulations on the respective RUL under macroscopic deformation.Further, we present an extension of the coupling methodology by developing strategies to obtain a macroscopically relaxed RUL. The RUL with Cauchy–Born boundaries is relaxed in a (nested) iterative procedure so that the macroscopic stress becomes zero for a certain equilibrium configuration. A stress–strain (i.e., membrane forces) response originating from macroscopically relaxed configuration is derived in two steps. The first step results in a stress-free configuration by finding a deformation map that corresponds to zero effective stress response. The subsequent step involves a multiplicative decomposition of the deformation and the pertinent push-forward of the effective stress

On the variationally consistent computational homogenization of elasticity in the incompressible limit

Background Computational homogenization is a well-established approach in material modeling with the purpose to account for strong micro-heterogeneity in an approximate fashion without excessive computational cost. However, the case of macroscopically incompressible response is still unresolved. Methods The computational framework for Variationally Consistent Homogenization (VCH) of (near) incompressible solids is discussed. A canonical formulation of the subscale problem, pertinent to a Representative Volume Element (RVE), is established, whereby complete macroscale incompressibility is obtained as the limit situation when all constituents are incompressible. Results Numerical results for single RVEs demonstrate the seamless character of the computational algorithm at the fully incompressible limit. Conclusions The suggested framework can seamlessly handle the transition from (macroscopically) compressible to incompressible response. The framework allows for the classical boundary conditions on the RVE as well as the generalized situation of weakly periodic boundary conditions

A poro-viscoelastic substitute model of fine-scale poroelasticity obtained from homogenization and numerical model reduction

Numerical model reduction is exploited for computational homogenization of the model problem of a poroelastic medium under transient conditions. It is assumed that the displacement and pore pressure fields possess macro-scale and sub-scale (fluctuation) parts. A linearly independent reduced basis is constructed for the sub-scale pressure field using POD. The corresponding reduced basis for the displacement field is constructed in the spirit of the NTFA strategy. Evolution equations that define an apparent poro-viscoelastic macro-scale model are obtained from the continuity equation pertinent to the RVE. The present model represents an extension of models available in literature in the sense that the pressure gradient is allowed to have a non-zero macro-scale component in the nested FE2 setting. The numerical results show excellent agreement between the results from numerical model reduction and direct numerical simulation. It was also shown that even 3D RVEs give tractable solution times for full-fledged FE2 computations

On the coupled thermo–electro–chemo–mechanical performance of structural batteries with emphasis on thermal effects

Carbon fibre (CF) based structural batteries is a type of battery designed to sustain mechanical loads. In this paper, a fully coupled thermo–electro–chemo–mechanical computational modelling framework for CF based structural batteries is presented. We consider the combined effects of lithium insertion in the carbon fibres leading to insertion strains, and thermal expansion/shrinkage of the constituents leading to thermal (free) strains, while assuming transverse isotropy. The numerical studies show that the developed framework is able to capture the coupled thermo–electro–chemo–mechanical behaviour. Moreover, it is found that the dominating source for heat generation during galvanostatic cycling is associated with discontinuities in the electrical and chemical potentials at the fibre/electrolyte interface. Further, a limited parameter study shows that the temperature change during electrochemical cycling is significantly influenced by the applied current, thermal properties of the constituents and heat exchange with the surroundings. Finally, for large temperature variations, e.g. as identified during relevant (dis)charge conditions, the magnitude of the thermal strains in the structural battery electrolyte (SBE) are found to be similar to the insertion induced strains

Homogenization of coupled flow and deformation in a porous material

In this paper we present a framework for computational homogenization of the fluid-solid interaction that pertains to the coupled deformation and flow of pore fluid in a fluid-saturated porous material. Large deformations are considered and the resulting problem is established in the material setting. In order to ensure a proper FE-mesh in the fluid domain of the RVE, we introduce a fictitious elastic solid in the pores; however, the adopted variational setting ensures that the fictitious material does not obscure the motion of the (physical) solid skeleton. For the subsequent numerical evaluation of the RVE-response, hyperelastic properties are assigned to the solid material, whereas the fluid motion is modeled as incompressible Stokes\u27 flow. Variationally consistent homogenization of the standard first order is adopted. The homogenization is selective in the sense that the resulting macroscale (upscaled) porous media model reminds about the classical one for a quasi-static problem with displacements and pore pressure as the unknown macroscale fields. Hence, the (relative) fluid velocity, i.e. seepage, "lives" only on the subscale and is part of the set of unknown fields in the RVE-problem. As to boundary conditions on the RVE, a mixture of Dirichlet and weakly periodic conditions is adopted. In the numerical examples, special attention is given to an evaluation of the Biot coefficient that occurs in classical phenomenological models for porous media

A 3D/2D comparison between heterogeneous mesoscale models of concrete

A model for 3D Statistical Volume Elements (SVEs) of mesoscale concrete is presented and employed in the context of computational homogenization. The model is based on voxelization where the SVE is subdivided into a number of voxels (cubes) which are treated as solid finite elements. The homogenized response is compared between 3D and 2D SVEs to study how the third spatial dimension influence the over-all results. The computational results show that the effective diffusivity of the 3D model is about 1.4 times that of the 2D model

On a volume averaged measure of macroscopic reinforcement slip in two-scale modeling of reinforced concrete

A two-scale model for reinforced concrete, in which the large-scale problem formulation is enriched by an effective reinforcement slip variable, is derived from the single-scale model describing the response of plain concrete, reinforcement steel, as well as the bond between them. The subscale problem on the Representative Volume Element (RVE) is correspondingly defined as finding the response of the RVE subjected to effective variables (strain, slip, and slip gradient) imposed from the large-scale.\ua0 A novel volumetric definition of effective reinforcement slip and its gradient is devised, and the corresponding subscale problem is formulated.\ua0 The newly-defined effective variables are imposed on the RVE in a weak sense via Lagrange multipliers. The response of the RVEs of different sizes was investigated by means of pull-through tests, and the novel boundary condition type was used in FE^2 analyses of a deep beam. Locally, prescribing the macroscopic reinforcement slip and its gradient in the proposed manner resulted in reduced RVE-size dependency of effective work conjugates, which allows for more objective description of reinforcement slip in two-scale modelling of reinforced concrete. Globally, this formulation produced more consistent amplitudes of effective slip fluctuations, as well as more consistent maximum crack width predictions

Mesoscale modelling of crack-induced diffusivity in concrete

Cracks have large impact on the diffusivity of concrete since they provide low-resistance pathways for moisture and chloride ions to migrate through the material. In this work, crack-induced diffusivity in concrete is modelled on the heterogeneous mesoscale and computationally homogenized to obtain macroscale diffusivity properties. Computations are carried out using the finite element method (FEM) on three-dimensional Statistical Volume Elements (SVEs) comprising the mesoscale constituents in terms of cement paste, aggregates and the Interfacial Transition Zone (ITZ). The SVEs are subjected to uni-axial tension loading and cracks are simulated by use of an isotropic damage model. In a damaged finite element, the crack plane is assumed to be perpendicular to the largest principle strain, and diffusivity properties are assigned to the element only in the in-plane direction of the crack by anisotropic constitutive modelling. The numerical results show that the macroscale diffusivity of concrete can be correlated to the applied mechanical straining of the SVE and that the macroscale diffusivity increases mainly in the transversal direction relative to the axis of imposed mechanical straining