A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global ClausiusPlanck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the twodimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurationalforce-based directional indicators. Here, successive crack propagation is performed by a staggered loadingrelease algorithm of energy minimization at frozen C. Miehe (B) · E. Gürses · M. Birkle crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi-phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation-driven context with non-trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non-trivial formulation of a traction-type boundary condition in the deformation-driven context. Modeling of FractureIn the context of fi nite elements there are different approaches for the modeling of discrete cracks. Two main approaches can be identifi ed, interelement discontinuities where cracks can run through the fi nite elements and intraelement discontinuities where cracks can run over the element boundaries. In our work we follow the intraelement approach which can be based on the insertion of interface elements between standard fi nite elements to model the ductile fracture or the removal of the connection between fi nite elements to model the brittle fracture. The interface elements may have a softening type material model relating tractions to displacment jumps to model cohesive fracture and in the limit the traction may drop directly to zero with the initiation of the crack yielding a brittle fracture response. Deformation Driven Homogenization of Nonlinear CompositesThe main aspects of the approach are governed by the incremental variational formulation for the local constitutive response as outlined in [1]. We extend the formulation to the solids with discontinuities by the defi nition of incremental potentials for the bulk and the crack surface seperately,in terms of free energies ψ b , ψ s and dissipation potentials φ b , φ s for the bulk and the crack surface, respectively. I and J stand for the set of internal variables that are computed by the minimization problem (??), F n+1 is the deformation gradient and δ n+1 is the displacement jump on the crack surface. As the key homogenization condition, we extend the minimization problem defi ned in [1] to a more general one considering the discontinuities,

The paper discusses formulations for the theoretical and numerical analysis of inelastic composites with scale separation. The basic underlying structure is a canonical variational setting in the fully nonlinear range based on incremental energy minimization. We focus on formulations of strain–driven homogenization for representative composite aggregates with emphasis on the development of canonical families of algorithms based on Lagrange and penalty functionals to cover alternative boundary constraints of (i.) linear deformations, (ii.) periodic deformations and (iii.) uniform tractions. As a key result, we present a compact matrix formulation for homogenization covering introduced alternative boundary constraints. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)