Andrey P. Jivkov

Review of pore network modelling of porous media: experimental characterisations, network constructions and applications to reactive transport

Pore network models have been applied widely for simulating a variety of different physical and chemical processes, including phase exchange, non-Newtonian displacement, non-Darcy flow, reactive transport and thermodynamically consistent oil layers. The realism of such modelling, i.e. the credibility of their predictions, depends to a large extent on the quality of the correspondence between the pore space of a given medium and the pore network constructed as its representation. The main experimental techniques for pore space characterisation, including direct imaging, mercury intrusion porosimetry and gas adsorption, are firstly summarised. A review of the main pore network construction techniques is then presented. Particular focus is given on how such constructions are adapted to the data from experimentally characterised pore systems. Current applications of pore network models are considered, with special emphasis on the effects of adsorption, dissolution and precipitation, as well as biomass growth, on transport coefficients. Pore network models are found to be a valuable tool for understanding and predicting meso-scale phenomena, linking single pore processes, where other techniques are more accurate, and the homogenised continuum porous media, used by engineering community.

Strain-driven corrosion crack growth: A pilot study of intergranular stress corrosion cracking

This work proposes a model for corrosion driven crack growth. The model poses a moving boundary problem, where a chemical attack removes material from the body. The rate of the chemical attack is a function of the strain along the body surface. No crack growth criterion is needed for the analysis. A finite strain formulation is used and the material model is assumed hyperelastic. The problem is stated for a large body, containing a large crack. A low frequency cyclic loading is considered. Thus, corrosion is assumed to dissolve material with a rate approximately proportional to the strain rate. The problem is solved using finite element method based program, enhanced with a procedure handling the moving boundary. Parametric studies are performed for a series of different initial shapes of the near-tip region. Presented results show that the crack growth rate is largely dependent on the initial crack geometry. For a set of initial shapes and load levels steady-state conditions of growth are achieved, while for the others the cracks show tendency to branch.

Meso-scale site-bond model for elasticity: theory and calibration

Abstract A meso-scale site-bond model is proposed to simulate the macroscopic elastic properties of isotropic materials. The microstructure of solids is represented by an assembly of truncated octahedral cells with sites at the cell centres and bonds linking the nearest neighbouring sites. Based on the equivalence of strain energy stored in a unit cell to strain energy stored in a continuum of identical volume, the normal and shear stiffness coefficients of bonds are derived from the given macroscopic elastic constants: Young’s modulus and Poisson’s ratio. To validate the obtained spring constants, benchmark tests including uniaxial tension and plane strain are performed. The simulated macroscopic elastic constants are in excellent agreement with the theoretical values. As a result, the proposed site-bond model can be used to simulate the macroscopic elastic behaviour of solids with Poisson’s ratios in the range from −1 up to 1/2.

Mesoscale Mechanical Model for Intergranular Stress Corrosion Cracking and Implications for Microstructure Engineering

The resistance of polycrystalline materials to intergranular cracking can be influenced by the microstructure. In sensitized stainless steels, for example, the grain boundaries prone to sensitization form paths of low resistance for intergranular stress corrosion cracking. The nonsensitized grain boundaries, such as twin boundaries, have been observed to encourage the formation of crack bridging ligaments. Computational models of intergranular cracking have been developed to investigate the consequences of crack bridging, through its effects on crack propagation in microstructures with different fractions of nonsensitized boundaries. This paper introduces the recently developed two-dimensional model for intergranular cracking with crack bridging, and reports its application to investigate the effect of grain size. It is shown that the size of the crack bridging zone depends on the grain size, and the shielding contribution depends on the relative size of the bridging zone compared to the crack length. It is concluded that both grain refinement and increase in the fraction of resistant boundaries can improve microstructure resistance to intergranular cracking. These observations are consistent with the effects of grain boundary engineering on stress corrosion cracking resistance in sensitized stainless steels.

Evolution of fatigue crack corrosion from surface irregularities

A moving boundary model is presented for crack nucleation and growth from surface flaws. It concerns with chemical attack that results in material dissolution. A controlling mechanism for evolution is the rupture of a brittle corrosion-protective film that is built up along the corroding surface. The evolution rate is a function of the degree of protective film damage caused by the surface straining. The problem is formulated for an elastic body containing a single and double pits. Low-frequency cyclic loading is considered. Numerical solution is proposed. The behaviours of a growing crack and of two competing cracks are described. Stages of incubation, blunting and steady-state growth characterise a single crack evolution. The steady-state growth rate is found independent of the initial geometry. Stages of independent growth, interactive growth and arrest of one crack characterise the evolution of two competing cracks. The lengths of the arrested cracks are presented as functions of the ratio between the pit depth for a series of different distances between the pits. It is emphasized that the solutions correspond to a homogeneous material. Further work is required to account for the material microstructure.

A Network Model for Diffusion in Media with Partially Resolvable Pore Space Characteristics

A microstructure-informed meso-scale model for diffusion of foreign species in porous media is proposed. The model is intended for media where the pore geometry data acquired experimentally represent a fraction of total porosity. A cellular complex, with a cell representing the average pore neighbourhood, is used to generate 3D graphs of sites at cell centres and bonds between neighbouring cells. The novel interpretation of pore systems as graphs allows for clear separation between topology (here connectedness) and physics (here diffusion) in the mathematical formulation of transport. Further, it allows for easy introduction of dynamics into the system, i.e., local changes in topology due to other physical mechanisms, such as micro-cracking or blockage of pores. A mapping between microstructure features and graph elements is used for model construction. The mapping is based on data for clays, where the experimentally resolved pore system comprises isolated elongated pores of preferred orientation with a large volume fraction of unresolved pores. Both the resolved and the “hidden” systems are accounted for. The graph geometry is described by a principal length, the cell size in the preferred orientation, and a secondary length, the cell size out of preferred orientation. This is considered as a representation of mineralogical heterogeneity of clays. Analysis on graphs, a specialisation of the discrete exterior calculus, is used to obtain connectivity and diffusivity properties of formed networks. Since the experimental data are not sufficient to determine the principal length, upper and lower limits are determined from the limited information. Effects of the principal cell size between limits and of the secondary cell size are studied. The results are within the range of experimentally measured macroscopic (bulk) diffusivity for the material studied, including anisotropic diffusion coefficients. The variation of calculated diffusivity coefficients with principal and secondary lengths provides an explanation for the variability in experimentally measured coefficients across different clays.

Discrete Lattice Model of Quasi-Brittle Fracture in Porous Graphite

Lattice models allow the incorporation of length scale dependent micro-structural features and damage mechanisms into analyses of the mechanical behaviour of materials. We describe our 3D lattice implementation and its use in fracture simulations. The method is particularly suitable for modelling fracture of nuclear graphite. This is a quasi-brittle material in which there is considerable non-linearity prior to final fracture due to the inherent porosity which triggers a field of local distributed failures upon mechanical and thermal loading. Microstructure representative models are generated with experimentally measured particle and pore size distributions and volume densities in two graphite grades. The results illustrate the effect of distributed porosity on the emerging stress-strain response and damage evolution. It is shown how the failure mode shifts from graceful, plastic-like, behaviour associated with substantial energy dissipation via distributed damage at lower porosities, to glass-like behaviour with negligible energy dissipation at higher porosities. Thus the work proposes a microstructure-informed methodology for integrity assessment of aging structures, where porosity increase is driven by environmental factors, such as radiation of nuclear graphite components.

Assessment of Local Approach Methods for Predicting End-of-Life Toughness of RPV Steels

Local approach methods are becoming increasingly popular as practical tools for cleavage fracture toughness prediction. Their application involves two distinct elements: calculation of ‘individual’ probabilities of failure, dictated by the local mechanical fields; and summation of these failure probabilities to predict the probability of component failure. In this work, we demonstrate that development of the local approach methods to date has been essentially focused on improving the criterion for predicting local failure as a function of the local mechanical fields. Yet, the existing methods fail to predict with sufficient accuracy the effects of irradiation and defect geometry on fracture toughness when the calculations are based on a common set of model parameters. A possible reason for this, common to all methods, is found in the calculation of the cumulative failure probability, which is based on the weakest-link argument. We discuss the implications of the weakest-link assumption, identify those situations where it needs to be reconsidered, and propose future work that will increase our understanding for improving the calculation of global failure probability.Copyright

A mathematical model for plasticity and damage

In this article we propose a discrete lattice model to simulate the elastic, plastic and failure behaviour of isotropic materials. Focus is given on the mathematical derivation of the lattice elements, nodes and edges, in the presence of plastic deformations and damage, i.e. stiffness degradation. By using discrete calculus and introducing non-local potential for plasticity, a force-based approach, we provide a matrix formulation necessary for software implementation. The output is a non-linear system with allowance for elasticity, plasticity and damage in lattices. This is the key tool for explicit analysis of micro-crack generation and population growth in plastically deforming metals, leading to macroscopic degradation of their mechanical properties and fitness for service. An illustrative example, analysing a local region of a node, is given to demonstrate the system performance.

Combined Numerical-Statistical Analyses of Damage and Failure of 2D and 3D Mesoscale Heterogeneous Concrete

Generation and packing algorithms are developed to create models of mesoscale heterogeneous concrete with randomly distributed elliptical/polygonal aggregates and circular/elliptical voids in two dimensions (2D) or ellipsoidal/polyhedral aggregates and spherical/ellipsoidal voids in three dimensions (3D). The generation process is based on the Monte Carlo simulation method wherein the aggregates and voids are generated from prescribed distributions of their size, shape, and volume fraction. A combined numerical-statistical method is proposed to investigate damage and failure of mesoscale heterogeneous concrete: the geometrical models are first generated and meshed automatically, simulated by using cohesive zone model, and then results are statistically analysed. Zero-thickness cohesive elements with different traction-separation laws within the mortar, within the aggregates, and at the interfaces between these phases are preinserted inside solid element meshes to represent potential cracks. The proposed methodology provides an effective and efficient tool for damage and failure analysis of mesoscale heterogeneous concrete, and a comprehensive study was conducted for both 2D and 3D concrete in this paper.