Multi-scale and multi-physics modelling for complex materials

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Evolving discontinuities and cohesive fracture

Multi-scale methods provide a new paradigm in many branches of sciences, including applied mechanics. However, at lower scales continuum mechanics can become less applicable, and more phenomena enter which involve discon- tinuities. The two main approaches to the modelling of discontinuities are briefly reviewed, followed by an in-depth discussion of cohesive models for fracture. In this discussion emphasis is put on a novel approach to incorporate triaxi- ality into cohesive-zone models, which enables for instance the modelling of crazing in polymers, or of splitting cracks in shear-critical concrete beams. This is followed by a discussion on the representation of cohesive crack models in a continuum format, where phase-field models seem promising

Evolving discontinuities in solids and structures

Evolving discontinuities can be modelled in a truly discrete sense, or in a smeared or continuum manner. Within the class of discrete models, cohesive-surface approaches are very versatile, in particular for heterogeneous materials. However, limitations exist, in particular related to stress triaxiality, which cannot be captured well in standard cohesive-surface models. We will therefore discuss an elegant enhancement of the cohesive-surface model to include stress triaxiality, which preserves the discrete character of cohesive-surface models. Subsequently, we will outline how the cohesive approach to fracture can be extended to multi-phase media, in particular fluid-saturated porous media. Whether a discontinuity is modelled via a continuum model, or in a discrete manner, advanced discretisation methods are needed to model the internal free boundary. Level sets, extended finite element methods and isogeometric analysis are important and promising tools in this respect. Examples will be given, including delamination in layered shells and fracture in fluid-saturated media. In smeared approaches to fracture higher-order spatial gradients typically evolve. Here, isogeometric analysis offers advantages by virtue of the smoothness of its basis functions, as will be demonstrated at the hand of a gradient-enhanced continuum damage model. In addition to approaches like NURBS that exploit tensor products for multi-dimensional generalisations, Powell-Sabin B-splines seem to be versatile, since they are defined on triangles, and thus share the ease and flexibility of mesh generation that characterises standard triangular finite elements. Another development in continuum approaches to fracture is the phase-field theory. We will discuss the formulation for brittle fracture, including its relation to gradient damage models, and extend the phase-field approach to cohesive fracture-surface models. We conclude with a concise discussion of the advantages of phase-field theories for damage and fracture

Gradient damage vs phase-field approaches for fracture: Similarities and differences

Gradient-enhanced damage models and phase-field models are seemingly very disparate approaches to fracture. Whereas gradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level, the phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, leading to the inclusion of spatial gradients as well. Herein, we will consider both approaches, and discuss their similarities and differences. The averaging (diffusion) equations for the averaging field and the phase-field will be compared, and it is shown that the diffusion equation for the phase-field can be conceived as a special case of the averaging equation of a gradient-damage model where the damage is averaged. Further, the role of the driving force is examined, and it is shown that subtle differences in the degradation functions commonly adopted in damage and phase-field approaches are key to the observation that, different from damage mechanics, the fracture process zone does not broaden in the wake of the crack tip

Powell–Sabin B-splines for smeared and discrete approaches to fracture in quasi-brittle materials

AbstractNon-Uniform Rational B-splines (NURBS) and T-splines can have some drawbacks when modelling damage and fracture. The use of Powell–Sabin B-splines, which are based on triangles, can by-pass these drawbacks. Herein, smeared as well as discrete approaches to fracture in quasi-brittle materials using Powell–Sabin B-splines are considered.For the smeared formulation, an implicit fourth-order gradient damage model is adopted. Since quadratic Powell–Sabin B-splines employ C1-continuous basis functions throughout the domain, they are well-suited for solving the fourth order partial differential equation that emerges in this higher order damage model. Moreover, they can be generated from an arbitrary triangulation without user intervention. Since Powell–Sabin B-splines are generated from a classical triangulation, they are not necessarily boundary-fitting and in that case they are not isogeometric in the strict sense.For discrete fracture approaches, the degree of continuity of T-splines is reduced to C0 at the crack tip. Hence, stresses need to be evaluated and weighted at the integration points in the vicinity of the crack tip in order to decide when the critical stress is reached. In practice, stress fields are highly irregular around crack tips. Furthermore, aligning a T-spline mesh with the new crack segment can be difficult. Powell–Sabin B-splines also remedy these drawbacks as they are C1-continuous at the crack tip and stresses can be directly computed, which vastly increases the accuracy and simplifies the implementation. Moreover, re-meshing is more straightforward using Powell–Sabin B-splines. A current limitation is that, in three dimensions, there is no procedure (yet) for constructing Powell–Sabin B-splines on arbitrary tetrahedral meshes

Fluid flow in fractured and fracturing porous media: A unified view

Fluid flow in fractures that pre-exist or propagate in a porous medium can have a major influence on the deformation and flow characteristics. With the aim of carrying out large-scale calculations at reasonable computing costs, a sub-grid scale model has been developed. While this model was originally embedded in extended finite element methods, thereby exploiting some special properties of the enrichment functions, we will herein show that, using proper micro-macro relations, in particular for the mass balance, sub-grid scale models can be coupled to a range of discretisation methods at the macroscopic scale, from standard interface elements to isogeometric finite element analysis