The origin of the energy split in phase‐field fracture and eigenfracture

The energy splits, which are used in phase‐field fracture and eigenfracture, are frequently motivated by the statement that crack nucleation and crack growth have to be prevented at compression loads. The intention of the article at hand is to show that the energy decomposition is an essential part of such fracture models, and to investigate the underlying problem and its physical interpretation. Finally, a general framework for the derivation of physically motivated energy splits is presented, which is not limited to certain bulk material models.


INTRODUCTION
The decomposition of the deformation energy potential in phase-field models for fracture can be traced back to Henry and Levine [1].Many further decompositions are proposed subsequently in the phase-field literature, for example, commonly used approaches are introduced in refs.[2] and [3].All those decompositions follow the format for the deformation energy potential  mech , where  is the phase-field field and () is the degradation function.The motivation for the energy split is to prevent unphysical crack nucleation and propagation at compression loads in early phase-field fracture models.Moreover, energy splits essentially control the entire crack nucleation and propagation process through the crack driving force ()   + , as well as the mechanical post-fracture behaviour through  − .However, a physical framework for the construction of energy splits has not been presented until the reinterpretation of Equation (1) in ref. [4].That is why, a variety of split models can be found in the literature.
The eigenfracture method has an intrinsic decomposition of the deformation energy potential in the form = { () for intact material ( eig = ) ( −  eig ) for cracked material , where the eigenstrain tensor  eig contains those deformations in the presence of a crack, which do not cause deformation energy.Thus, the decomposition in eigenfracture is controlled through the eigenstrain tensor.Although Schmidt et al. [5] have given certain conditions for the eigenstrain tensor in their Γ-convergence proof, this is widely ignored subsequently.Instead, the splits from phase-field fracture are adopted as Hence, the split models have become an essential part of regularised fracture models.However, an increasing number of publications have reported unphysical predictions for the mechanical post-fracture behaviour and, as a consequence, for the crack driving force, for example, in May et al. [6] and Strobl and Seelig [7].These discrepancies have motivated the authors to investigate the mathematical origin of the energy decomposition and the physical relation with the deformation kinematics at cracks.Finally, a generalised framework for the derivation of the energy decompositions could be developed by means of Representative Crack Elements (RCE) and computational homogenisation.

THE FREE DISCONTINUITY PROBLEM
Regularised fracture models refer back to Griffith's consideration on the energetic transformation during crack propagation.Then, the released elastic energy of a body with fixed Dirichlet boundary conditions is in balance with the dissipated energy to form new surfaces, which can be written in the form where  is the domain of the body, Γ ⊆  the domain of the crack,  c the fracture toughness and  the displacement field with the weak derivative .This variational problem belongs to the free discontinuity problems, in which the unknown field  can exhibit discontinuities at the subdomain Γ.However, the size and location of Γ is also unknown (free).It has been turned out that the numerical calculation of solutions for the free discontinuity problem is challenging.One difficult aspect for the discretisation in numerical solution schemes is that the unknown and variable domain of the discontinuity bounds the integration domains in Equation (5).Thus, the discretisation needs to be (adaptively) adjusted to the integration domains, that is, to the current shape and location of the crack.However, adaptive discretisation schemes have to map the deformation history (internal variables) towards the modified mesh.A general procedure for these mappings, which can be applied to all material models, does not exist.
Regularised formulations for the domain Γ have been proposed as an alternative to the variable integration domains with discrete bounds at the discontinuities.Well known regularised formulations for the free discontinuity problem of brittle fracture are for instance phase-field for fracture and eigenfracture where  is the regularisation parameter and   eig ≠0 the domain of the crack neighbourhood, compare Figure 1.
In both formulations, the deformation energy potential is expressed through an integration on the entire domain , which also includes the crack Γ.On the one side, the discretisation does not need to be adjusted nor to follow the cracks, which allows to apply common schemes of computational mechanics, like the Finite Element Method.On the other side, the deformation energy potential is not exclusively evaluated at points of solid material without cracks, compare Figure 2A.Hence, it is further evaluated at location, where the crack is present in Equations ( 6) and (7).At those cracked material points, the derivative of (discontinuous) displacement field does not represent the deformation of the solid material nearby the crack, because it also include a portion caused by the relative displacements between the crack surfaces, see Figure 2A.
The discontinuous functions of the free discontinuity problem can be described in the space of Special functions of Bounded Variation (SBV).The properties of this space are intensively studied, for example, by De Giorgi et al. [8] and Ambrosio et al. [9].It is shown that the weak derivative  at the discontinuity additively decomposes into an absolutely continuous part  a  and a jump part  j   =  a  +  j .
The jump part is equipped with the structure where ⟦⟧ is the displacement jump vector and  is the normal vector at the discontinuity.The physical interpretation of this decomposition in the context of brittle fracture relates the absolutely continuous part  a  to the solid deformation and the jump part  j  to the contribution of the crack, compare Figure 2A.Then, the free discontinuity problem of Equation ( 5) can be rewritten as F I G U R E 3 (A) A possible realisation for an representative crack element and (B) its deformed state.
The comparison of this formulation to the regularised formulations in Equations ( 6) and (7) shows that the used energy decompositions from Equations ( 1) and ( 4) are realisations of the decomposition of the weak derivative.Thus, the energy decompositions of phase-field fracture and eigenfracture can be equivalently transformed to and where  solid =  solid () =  solid ( a ) is the material behaviour without a crack and  crack =  crack ( a ) is the (unknown) material behaviour in the presence of a crack.Consequently, the energy decomposition consists of two physically clearly defined states for a given deformation : • intact material  solid (), which can be described by any conventional material model, and • broken/cracked material  crack , which needs to be determined.
For the case that the decomposition of  towards  a  is known at each point on , the same conventional material model could be used in the form  crack =  solid ( a ).In general, the determination of  crack is the missing step to solve the energy decomposition in the form of Equations ( 11) or (12), which is the purpose of the framework of RCE.

REPRESENTATIVE CRACK ELEMENTS
The basic concept for the RCE is adopted from computational homogenisation.Let us consider a small portion of the crack and the solid material adjacent to both crack surfaces and call it RCE.Then, a numerical model of the RCE consists of solid material and a portion of a crack, compare Figure 3.The application of standard homogenisation relations allows to couple the RCE to  crack of the regularised fracture model above.Following some standard arguments in computational homogenisation [10], the crack portion in the RCE can be considered to be plane and the deformations in the adjacent solid portions to be homogeneous, if the RCE is sufficiently small.Adopting the common kinematic coupling operators yields the relation  8) and ( 9) allows the conclusion that the chosen assumptions for the RCE allows to reproduce the desired structure for the displacement derivative and the form of the jump part  j .Thereby, the shape and size of the RCE has no general limitations.
Finally, the RCE can be treated as an ordinary problem of solid mechanics with contact at the crack surface.The same conventional material model as applied for  solid is applied to the solid regions of the RCE.The solution of the problem is very cheap, because the entire problem can be formulated w.r.t. the unknown jump of the displacement field ⟦ ū⟧.The solution for  crack and its derivatives for the regularised fracture model reads Efficient numerical methods for the calculation of the homogenised stress and material tangent for the RCE are presented in ref. [11].
Next to the decomposition of the displacement derivative and the deformation energy  crack for solid material in the presence of a crack, the RCE model can be straightforwardly extended to processes at the crack surfaces and inside the crack gap as presented in the subsequent examples.

SELECTED APPLICATIONS OF THE RCE FRAMEWORK
A deformable sliding block is presented by Oden and Pires [12] and Simo and Laursen [13] and simulated by the FEM with different contact models.The block is in frictional contact with the rigid foundation, modelled by Coulombs friction approach.While pulled from the side a normal compression is acting on top of the block.This mechanical problem is adopted and modelled by the eigenfracture method and the FEM.An RCE with frictional contact is formulated as presented in ref. [11].The stress distribution at the interface is plotted in Figure 4B together with the results of the conventional contact models from refs.[12] and [13].All three results matches relatively well.Moreover, the transition point between frictional sticking and frictional sliding at about  = 3000 mm is predicted accurately.
The RCE framework is also applied to multi-physics by means of phase-field fracture in the context of the FEM, like thermo-mechanics at the thermal shock of a ceramic disc [14] or electro-mechanics of electro-active polymers [15].Like crack surface friction, further processes in the crack gap can be considered in these applications, too.The benchmark example of a pre-crack plate applied to an electro-mechanical RCE formulation allows to study the influence of different permeability assumptions for the crack gap on the electrical behaviour in the neighbourhood of the crack, compare Figure 5A.
While permeability is often assumed for closed cracks, different approaches are used for opened cracks in literature, for example, impermeable, permeable and semi-permeable.An energetically consistent formulation for a permeable medium and corresponding tractions at the crack surfaces is shown in ref. [16].Such crack-face constraints can be directly modelled in the RCE.The natural constitutive conditions are listed in the following for the common cases mentioned above: The RCE allows consideration of linear and nonlinear crack-face conditions without modifications to the phase-field formulation.The electrical potential of the plate is presented in Figure 5B.The realistic prediction of the electrical behaviour in the gap is of interest because it directly influences the electrical potential in the neighbourhood of the crack tip.The notched tensile test at different applied electrical potentials in Figure 6 demonstrates the significant influence on the peak force and, thus, on the crack propagation.The simulation is performed in a finite strain framework using Ogden's hyperelastic approach and the Extended Tube model.The material parameters are given in Table 1.The state of the specimen at the peak load is presented in Figure 7.The electro-mechanical RCE model also predicts the discontinuity of the electrical potential at the crack.This interaction is frequently ignored in electro-mechanical fracture models.The determination of the crack normal for general materials are a major challenge of the RCE framework.In both applications above, the maximum principle stress criterion is applied as introduced in ref. [4].However, it is known from the experimental findings of Rozen-Levy et al. [17] that the direction of maximal fracture energy is more realistic for anisotropic and heterogeneous materials.The solution of this challenging optimisation problem is studied in ref. [18] for simple examples.A generalised, robust and efficient solution scheme for the energetically motivated crack normal is part of ongoing research activities.

F
I G U R E 1 (A) Profile of the phase-field field through a crack and (B) the crack neighbourhood definition in eigenfracture.F I G U R E 2 (A) Decomposition of the weak derivative  and (B) the crack neighbourhood definition in eigenfracture.

F I G U R E 4 1 𝑙 1 ⟦
Frictional contact of a frictional sliding block on a rigid ground by means of eigenfracture, (A) discretisation and pre-eroded elements, (B) comparison of the interface stress distribution to those of contact models from literature. at the RCE.The displacement derivative at a material point of the fracture model (phase-field or eigenfracture) |  decomposes into the deformation of the solid H and the contribution of the crack ū⟧ ⊗  1 .The comparison to Equations (

F I G U R E 5
Pre-cracked plate, (A) system and boundary conditions, (B) electrical potential through the crack.F I G U R E 6 Tension test, (A) sketch and boundary conditions, (B) critical failure load.