Computational homogenisation of phase-field fracture

In this manuscript, the computational homogenisation of phase-field fractures is addressed. To this end, a variationally consistent two-scale phase-field fracture framework is developed, which formulates the coupled momentum balance and phase-field evolution equations at the macro-scale as well as at the Representative Volume Element (RVE) scale. The phase-field variable represent fractures at the RVE scale, however, at the macro-scale, it is treated as an auxiliary variable. The latter interpretation follows from the homogenisation of the phase-field through volume or a surface-average. For either homogenisation choices, the set of macro-scale and sub-scale equations, and the pertinent macro-homogeneity satisfying boundary conditions are established. As a special case, the concept of selective homogenisation is introduced, where the phase-field is chosen to live only in the RVE domain, thereby eliminating the macro-scale phase-field evolution equation. Numerical experiments demonstrate the local macro-scale material behaviour of the selective homogenisation based two-scale phase-field fracture model, while its non-selective counterpart yields a non-local macro-scale material behaviour

Computational aspects of the weak micro‐periodicity saddle point problem

The finite element implementation of the weak micro-periodicity problem in computational homogenisation requires special preconditioning techniques owing to the saddle point formulation. The saddle point nature arises from enforcing periodicity constraints using Lagrange multipliers. This manuscript addresses the solution techniques and preconditioning options for the aforementioned problem in a monolithic setting. Furthermore, an alternative technique is proposed, based on a linear multi-point constraints strategy. The latter approach eliminates the Lagrange multiplier Degrees of Freedom (DOFs), thereby preventing the break-down of conventional incomplete LU (ILU) variants and multi-grid method based preconditioners

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

Diffuse interface modeling and variationally consistent homogenization of fluid transport in fractured porous media

We critically assess diffuse interface models for fluid transport in fractured porous media. Such models, often called fracture phase field models, are commonly used to simulate hydraulic stimulation or hydraulic fracturing of fluid-saturated porous rock. In this paper, we focus on the less complex case of fluid transport in stationary fracture networks that is triggered by a hydro-mechanical interaction of the fluid in the fractures with a surrounding poroelastic matrix material. In other words, fracture propagation is not taken into account. This allows us to validate the diffuse interface model quantitatively and to benchmark it against solutions obtained from sharp interface formulations and analytical solutions. We introduce the relevant equations for the sharp and diffuse, i.e. fracture phase field, interface formulations. Moreover, we derive the scale-transition rules for upscaling the fluid-transport problem towards a viscoelastic substitute model via Variationally Consistent Computational Homogenization. This allows us to measure the attenuation associated with fluid transport on the sub-scale. From the numerical investigations we conclude that the conventional diffuse interface formulation fails in predicting the fluid-transport behavior appropriately. The results even tend to be non-physical under certain conditions. We, therefore, propose a modification of the interpolation functions used in the diffuse interface model that leads to reasonable results and to a good approximation of the reference solutions

Combining spectral and POD modes to improve error estimation of numerical model reduction for porous media

Numerical model reduction (NMR) is used to solve the microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. The reduction technique and an associated error estimator for the NMR error have been presented in prior work, where both spectral decomposition (SD) and proper orthogonal decomposition (POD) were used to construct the reduced basis. It was shown that the POD basis performs better w.r.t. minimizing the residual, but the SD basis has some advantageous properties for the estimator. Since it is the estimated error that will govern the error control, the most efficient procedure is the one that results in the lowest error bound. The main contribution of this paper is further development of the previous work with a proposed combined basis constructed using both SD and POD modes together with an adaptive mode selection strategy. The performance of the combined basis is compared to (i) the pure SD basis and (ii) the pure POD basis via numerical examples. The examples show that it is possible to find a combination of SD/POD modes which is improved, i.e. it yields a smaller estimate, compared to the cases of pure SD or pure POD

A posteriori error estimation for numerical model reduction in computational homogenization of porous media

Numerical model reduction is adopted for solving the microscale problem that arizes from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. A reduced basis is obtained for the pressure field using (i) spectral decomposition (SD) and (ii) proper orthogonal decomposition (POD). This strategy has been used in previous work—the main contribution of this article is the extension with an a posteriori estimator for assessing the error in (i) energy norm and in (ii) a given quantity of interest. The error estimator builds on previous work by the authors; the novelty presented in this article is the generalization of the estimator to a coupled problem, and, more importantly, to accommodate the estimator for a POD basis rather than the SD basis. Guaranteed, fully computable and low-cost bounds are derived and the performance of the error estimates is demonstrated via numerical results

Modeling and computational homogenization of chloride diffusion in three-phase meso-scale concrete

A computational homogenization technique for modeling diffusion in concrete is introduced with emphasis on the influence of the aggregate content and variability. The highly heterogeneous material is investigated on different scales by combining Variationally Consistent Homogenization on numerical microstructures with analytical techniques accounting for lower, unresolved, length scales. The concrete structure consists of the cement paste, the embedded aggregates, and the Interfacial Transition Zone (ITZ) in between the two. Diffusion takes place in the cement phase, as well as in the ITZ. Since the thickness of the ITZ is, typically, much smaller than the diameter of the aggregates, the effect of the ITZ can be modelled as a surface transport around the aggregates. The occurrence of different aggregate sizes is described via the Particle Size Distribution for given sieve curves, as described in design codes. The Particle Size Distribution curve is split into two parts. The effect of smaller aggregates is homogenized analytically using a mixture rule. This results in an effective matrix material consisting of cement paste and the smaller aggregates. Synthetic structures are then generated numerically to account for the larger aggregates. At first, a dense sphere packing is created based on the Particle Size Distribution. This information is used to generate a weighted Voronoi diagram, which is modified by a shrinking process. This procedure allows us to create periodic Representative Volume Elements for numerical investigations. The overall diffusivity of the concrete mixture is evaluated upon using Variationally Consistent Homogenization, in the context of Finite Element analysis, for the generated RVEs and compared with analytical homogenization results and experimental data. It is found that, depending on the Particle Size Distribution, the ITZ has a large effect on the effective properties

Variationally consistent modeling of a sensor-actuator based on shape-morphing from electro-chemical–mechanical interactions

This paper concerns the computational modeling of a class of carbon fiber composites, known as shape-morphing and strain-sensing composites. The actuating and sensing performance of such (smart) materials is achieved by the interplay between electrochemistry and mechanics, in particular the ability of carbon fibers to (de)intercalate Li-ions repeatedly. We focus on the actuation and sensing properties of a beam in conjunction with the appropriate “through-the-thickness” properties. Thus, the electro-chemo-mechanical analysis is essentially two-dimensional, and it is possible to rely heavily on the results in Carlstedt et al. (2020). More specifically, the cross-sectional design is composed of two electrodes, consisting of (partly) lithiated carbon fibers embedded in structural battery electrolyte (SBE), on either side of a separator. As a result, the modeling is hierarchical in the sense that (macroscale) beam action is combined with electro-chemo-mechanical interaction along the beam. The setup is able to work as sensor or actuator depending on the choice of control (and response) variables. Although quite idealized, this design allows for a qualitative investigation. In this paper we demonstrate the capability of the developed framework to simulate both the actuator and sensor modes. As proof of concept, we show that both modes of functionality can be captured using the developed framework. For the actuator mode, the predicted deformation is found to be in close agreement with experimental data. Further, the sensor-mode is found to agree with experimental data available in the literature

Finite Element Simulation of the Performance of a Structural Electrolyte

This contribution concerns the multi-scale and multi-physics finite element analysis of structural power composites, i.e. multifunctional composites with simultaneous load bearing and energy storing functionality. We are particularly interested in obtaining the effective macro-scale properties of the structural electrolyte by employing computational homogenization to capture the effects of micro-heterogeneities on the sub-scale. The sub-scale problem is defined by a statistical volume element that is numerically generated, and the effective properties are obtained by conducting virtual material testing on the synthetic microstructure

Computational modelling of structural batteries accounting for stress-assisted convection in the electrolyte

Structural batteries consist of carbon fibres embedded in a porous structural battery electrolyte (SBE), which is composed of two continuous phases: a solid polymer skeleton and a liquid electrolyte containing Li-salt. In this paper we elaborate on a computational modelling framework to study the electro-chemo-mechanical properties of such structural batteries while accounting for the combined action from migration as well as stress-assisted diffusion and convection in the electrolyte. Further, we consider effects of lithium insertion in the carbon fibres, leading to insertion strains. The focus is placed on how the convective contribution to the mass transport within the SBE affects the general electro-chemo-mechanical properties. The numerical results indicate that the convective contribution has only minor influence on the multifunctional performance when the mechanical loading is caused by constrained deformation of constituents during electro-chemical cycling. However, in the case of externally applied mechanical loading that causes severe deformation of the SBE, or when large current pulses are applied, the convective contribution has noticeable influence on the electro-chemical performance. In addition, it is shown that the porosity of the SBE, which affects the effective stiffness as well as the mobility and permeability, has significant influence on the combined mechanical and electro-chemical performance