Phase‐field Fracture with Representative Crack Elements for Non‐linear Material Behaviour

The mechanical energy potential of phase‐field fracture models is subdivided into a portion which (actively) drives the crack and a passive portion. This decompositions depends further on the crack state (opened, closed) in order to consider the re‐contact of the crack surfaces. The identification of the crack state and the decomposition is mostly approximated based on splits of the deformation or stress tensor. Stobel and Seelig [1], and Steinke and Kaliske [2] have shown unrealistic predictions for the crack kinematic for those models in quasi‐static and dynamic analyses. The approach proposed by these authors allows to predict the crack kinematic consistently. Nevertheless, this model is restricted to linear, isotropic elasticity and small deformations.


Introduction
The framework of Representative Crack Elements (RCE) for phase-field fracture is used to derive regularised crack models where the crack kinematics follows those of a discrete crack model with contact. Therefore, a representative part of discrete crack is coupled to the phase-field model by means of computational homogenisation. This approach can be formulated in a fully variational framework and avoids artificial definitions for crack contact and the crack driving force.
The RCE framework for phase-field fracture allows to derive crack models, which successfully represent the crack kinematics of discrete cracks, and can, therefore, replace discrete crack models. This property is shown to be missing in the phase-field formulations with artificial splits so far, e.g. in Strobl and Seelig [1], Schlüter [4] and Steinke and Kaliske [2]. Their important findings are ignored in further developments in phase-field fracture, which have yield to numerous models based on insufficient contact and crack driving formulations (splits). The crack propagation at a shear plate with different models for the initial crack in Fig. 1 demonstrates the importance of realistic crack kinematics. a) b) c) Fig. 1: Comparison of crack propagation at a single-edge notch specimen at shear load with a) a discrete initial crack, b) a phase-field initial crack applying phase-field fracture with spectral split [5] and c) a phase-field initial crack applying phase-field fracture derived from RCE framework [3].
The extension of the variational RCE framework towards non-linear bulk material is presented in the following. The basic relations of computational homogenisation applied to phase-field fracture are introduced. The RCE problem for non-linear constitutives is formulated, linearised and an iterative solution scheme derived.

Application of Computational Homogenisation to Phase-field Fracture
The variational formulation for continuum solid mechanics can be given as Principle of Virtual Power. The total virtual power balances the virtual power of internal stresses and external forces. This first variation has to vanish for any admissible field of generalised virtual displacements δv and the corresponding generalised virtual strains D(δv) For quasi-static phase-field fracture, the generalised displacements consist of the mechanical displacement vector and the phase- The generalised strains for small deformations are given by the linearised strain tensor, the phase-field variable and its gradient The dual products of generalised displacements and strains with the generalised external forces f and internal stresses Σ form the power terms for the global minimisation task. The stresses follow the constitutive laws given in the form of a HELMHOLTZ free energy potential as where α are the internal variables. In the following, parts of the constitutive behaviour are derived from a representative crack element by means of homogenisation. The RCE forms a classical, variational problem in the framework of CAUCHY-BOLTZMANN mechanics, wherē Both variational problems are coupled by the following kinematical insertion and homogenisation operators The thermo-dynamically conjugated internal stresses and external forces of the deformed RCE are a direct result of the Principle of Multiscale Virtual Power [6], which balances the total virtual power of the two coupled variational problems.

Concept of Representative Crack Elements (RCE)
The constitutive behaviour for phase-field fracture is postulated as superposition of those of intact material ψ 0 and fully degraded (cracked) material ψ c The quite simple model given in Fig. 2 is proposed to represent the deformation kinematics at a crack, i.e. ψ c . Two blocks of homogeneous material (B 1 ,B 2 ) and equal size are separated by a discrete crackB Γ , where crack surface contact is under consideration. A CARTESIAN coordinate system (N 1 , N 2 , N 3 ) is introduced for the RCE. The first axis N 1 coincides with the normal direction of the crack surface. Homogeneous deformations are postulated in the subdomains of the RCE. The strain through the crack follows from the kinematic compatibility at the boundaries of the subdomains, where l Γ is the initial thickness of the crack and ∆ū Γ the displacement discontinuity. A limit analysis for l Γ → 0 via L'HOSPITAL's rule yields and the displacement field This displacement field is defined and linear in terms of u| x , E| x and Γ i . The first two terms are prescribed by the phase-field problem to the RCE model via DIRICHLET conditions, whereas the crack deformations Γ i are unknown. The linearisation and discretisation of this minimisation task for the RCE, compare Eq. (1), yield the iterative NEWTON-RAPHSON scheme where Then, the stress, the internal variables and material tangent for the phase-field formulations read www.gamm-proceedings.com

Applications
The usage of common material models given in implicit or explicit formulations for the bulk material of at the RCE is straightforward and do not require modifications or a special treatment. Basic properties and kinematical consistency are demonstrated at a pre-cracked compression plate shown in Fig. 3. Linear visco-elasticity, compare [7], results in constant material tangents for constant time increments. Therefore, those problems can be covered by the linear RCE framework in [3]. However, the concept of thermo-dynamics with internal variables for irreversible and dissipative processes is first addressed in the formulation above. The compressive displacement and its release (starting at 60 % and ending at 70 % of the time) cause viscous deformations and, thus, crack opening during unloading as well as delayed crack closing. Both effects are correctly predicted by the simulation shown in Fig. 4.  For the bulk material using MOHR-COULOMB elasto-plasticity with isotropic hardening, the permanent deformation induced during compression causes an early crack opening. This can be seen in Fig. 5 comparing the normal reaction force to those of a pure elastic simulation, where the compression is fully released after 80 % of time.
The presented framework allows also to handle further deformation mechanisms which directly act at the crack surface, e.g. friction, corrosion etc. Crack surface friction of COULOMB type is considered at a pre-cracked plate applied to tension and shear strains by means of periodic boundary conditions. The normal and tangential reaction force is shown in Fig. 6 for different friction coefficients. Sticking and sliding states can be clearly distinguished. The corresponding deformation states of the compressed plate, at the end of the sticking state and during the plate sliding are shown in Fig. 6b.