Phase-field computation of brittle fracture: robustness, efficiency, and characterisation of solution non-uniqueness

点击翻译按钮获取中文摘要

Modelling and accurate prediction of complicated fracture processes in elastic solids including crack initiation, propagation, branching, and coalescence, is a very challenging topic of engineering interest. We focus on brittle materials where fracture occurs prior to significant plastic (permanent) deformation.The phase-field approach enables realistic simulations of such material failure processes by associating a continuous field (the crack phase-field) to the state of the material, ranging from intact to fully broken. Models of material softening may result in mathematically non-unique solutions. Their probability of appearance is in theory determined by energetic relations, and in the real material by random inhomogeneities.In deterministic computations the following difficulties appear: sharp cracks are modelled by a steep variation of the phase-field, necessitating very fine meshes in numerical discretisations which may be computationally very expensive. To deal with this via adaptive meshing, the development of efficient and reliable error estimators is required. As damage evolution and fracture are irreversible processes, the solution evolves in time, computationally in discrete time-steps. The time-step choice strongly influences the solution process and the possible appearance of non-uniqueness. Error indicators will be developed to control the time-step selection as regards algorithmic efficiency and robustness. In each time-step, a possibly non-convex energy functional has to be minimised displaying multiple local minima and hence causing problems for numerical algorithms. Existing partitioned and monolithic approaches will be further advanced as regards the robustness and efficiency. Particular attention will be paid to the capture of all local minima.To capture the non-unique nature of solutions, the probability of their appearance (determined by their energetic relations) will be described by random fields. This non-uniqueness computationally appears with different mesh-refinements, time-stepping algorithms, and within the minimisation algorithm in each step. All of these will be parameterised locally by appropriate random variables, thus capturing the evolution of all possible solutions via random fields.The description of the crack propagation process will model both the displacement and phase-field in each point in space and each instant in time as a random variable, which then is a spatio-temporal random field. This describes the different probabilities of the state of the material, i.e. probability of crack development and the associated displacement.Such an approach generates a large number of parameters which describe the solutions and their evolution. Numerically sparse techniques for dealing with high-dimensional parametric problems will be further developed to suit the phase-field approach.