Fatigue fracture simulations with the phase‐field method (PFM) lack efficiency due to the fine meshes required, especially when each load cycle is simulated explicitly. Recent developments in combining the phase‐field method for brittle fracture with the extended/generalised finite element method (XFEM/GFEM) show a remarkable reduction of the number of degrees of freedom and thus a reduction of computational effort while retaining or even improving accuracy. In this paper, the combined extended phase‐field method (XPFM) is expanded to fatigue processes in a two‐dimensional setting. A new stabilisation of the phase‐field transformation function is proposed and a convergence criterion for the nested staggered solution process and enrichment scheme update procedure as well as an adaptive integration technique for the non‐polynomial ansatz functions is incorporated. Several numerical examples emphasise the benefits and the efficiency of the newly developed method.

Fine-grained high-strength concrete has already been tested extensively regarding its uniaxial strength. However, there is a lack of research on the multiaxial performance. In this contribution, some biaxial tests are investigated in order to compare the multiaxial load-bearing behaviour of fine-grained concretes with that of high-strength concretes with normal aggregate from the literature. The comparison pertains to the general biaxial load-bearing behaviour of concrete, the applicability of already existing fracture criteria and the extrapolation for the numerical investigation. This provides an insight into the applicability of existing data for the material characterisation of this fine-grained concrete and, in particular, to compensate for the lack of investigations on fine-grained concretes in general. It is shown, that the calibration of material models for fine-grained concretes based on literature results or normal-grained concrete with similar strength capacity is possible, as long as the uniaxial strength values and the modulus of elasticity are known. For the numerical simulation, a Microplane Drucker–Prager cap plasticity model is introduced and fitted in the first step to the biaxial compression tests. The model parameters are set into relation with the macroscopic quantities, gained from the observable behaviour of the concrete under uniaxial and biaxial compressive loading. It is shown that the model is able to capture the yielding and hardening effects of fine-grained high-strength concrete in different directions.

Complex crack processes like crack initiation, branching, and coalescence can be simulated with the phase-field method for fracture. Here, the criterion for crack propagation is considered within the formulation of the partial differential equation. Its solution, the scalar phase-field, indicates the crack in a smeared manner. This makes the evaluation of a separate criterion for crack propagation and the tracking of the crack geometry obsolete. A challenge for the phase-field method for fracture, however, is the accurate and efficient reproduction of the phase-field, the displacement field, and their gradients. For this, the use of extremely fine meshes is often required within the standard phase-field method. In this contribution, an enhancement of the phase-field method through the adjustment of the ansatz functions is presented. A transformation of the phase-field ansatz by embedding a quadratic ansatz into an exponential function is utilised to improve the approximation of the phase-field even with coarse meshes. The crack is not additionally restricted by the ansatz and can still develop independently of the mesh geometry. Furthermore, the displacement ansatz is extended by a term with adjusted shape-functions carrying information about the crack geometry from the phase-field. It can reproduce the high displacement gradients across the crack. The modified shape-functions are calculated for each enriched element on a submesh by considering the local reduction of stiffness due to the phase-field. Since these shape-functions are directly coupled to the phase-field ansatz, no additional discretisation of the crack geometry is required.