No abstract

SUMMARYThis work develops variational principles for the coupled problem of standard and extended Cahn–Hilliard‐type species diffusion in solids undergoing finite elastic deformations. It shows that the coupled problem of diffusion in deforming solids, accounting for phenomena like swelling, diffusion‐induced stress generation and possible phase segregation caused by the diffusing species, is related to an intrinsic mixed variational principle. It determines the rates of deformation and concentration along with the chemical potential, where the latter plays the role of a mixed variable. The principle characterizes a canonically compact model structure, where the three governing equations involved, that is, the mechanical equilibrium condition, the mass balance for the species content and a microforce balance that determines the chemical potential, appear as the Euler equation of a variational statement. The existence of the variational principle underlines an inherent symmetry of the coupled deformation–diffusion problem. This can be exploited in the numerical implementation by the construction of time‐discrete and space‐discrete incremental potentials, which fully determine the update problems of typical time stepping procedures. The mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation–diffusion problem based on low‐order basis functions. They induce in a natural format the choice of symmetric solvers for Newton‐type iterative updates, providing a speedup and reduction of data storage when compared with non‐symmetric implementations. This is a strong argument for the use of the developed variational principles in the computational design of deformation–diffusion problems. Copyright © 2014 John Wiley & Sons, Ltd.

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1][2][3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4] . To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. subject to the Dirichlet constraints W Γ(t) = {d | d(x, t) = 1 at x ∈ Γ(t)}. The crack functional Γ l in terms of the crack surface density function γ(d, ∇d) governs the regularization by the length scale parameter l giving for l → 0 the sharp crack topology. The reader is referred to the recent work [1] for a more detailed derivation of the diffusive crack topology.

The numerical modeling of failure mechanisms in plates and shells due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. In this paper, we extend ideas recently outlined in [1,2] towards the phase field modeling of fracture in dimension-reduced continua with application to Kirchhoff plates and shells. The introduction of history fields, containing the maximum reference energy obtained in history, provides a very transparent representation of the coupled balance equations and allows the construction of an extremely robust operator split technique. The performance of the proposed models is demonstrated by means of representative numerical examples. Approximation of Crack Topology by Phase FieldWe develop a crack line theory where the crack is considered as a line on the reference surface of the thin-walled structure. To this end, we introduce the time-dependent crack phase field d ∈ [0, 1] characterizing for d = 0 the unbroken and for d = 1 the fully broken state of the material. A regularized crack phase field d(x, t) is obtained from the minimization principleThe crack functional L in terms of the crack line density function γ(d, ∇d) governs the regularization by the length scale parameter l giving for l → 0 the sharp crack topology. Incremental Variational Principle of Diffusive Fracture in PlatesThe basic fields of the coupled problem are the deflection field w and the fracture phase field d. The global energy storage functional considering standard linear theory of elasticity for plates readswhere we introduced the reference energy of the Kirchhoff plate ψ 0,b in terms of the curvature κ(w)and K := EtThe degradation function g(d) = (1 − d) 2 describes the degradation of ψ 0,b with evolving damage. The rate of the energy functional is governed by the bending moment m := ∂ κ ψ, the energetic force f := −∂ d ψ, and the micro force χ :We propose an extended dissipation functional in terms of β dual to d, the Lagrange parameter λ, and the threshold function t c for a rate independent class of dissipation functions satisfying the non-reversible evolution of the crack phase fieldThe rate potential functional Π λ = E + D λ − P balances the internal power E + D λ with the power due to external loading P . Now, the governing equations are obtained from the argument of virtual power that we base on the variational statement {ẇ,ḋ, β, λ} = Arg{ staṫ w,ḋ,β,λ≥0Variation of the functional with respect to the four field variables gives the coupled Euler-Lagrange equations

Recently developed continuum phase field models for brittle fracture show excellent modeling capability in situations with complex crack topologies including branching in the small and large strain applications. This work presents a generalization towards fully coupled multi-physics problems at large strains. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi field material response, where the focus is put on the model problem of finite thermoelasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress-based criteria, the constitutive modeling of degradation of non-mechanical fluxes on generated crack faces. Particular assumptions are made on the generation of convective heat exchanges approximating surface load integrals of the sharp crack approach by distinct volume integrals. The coupling effect is also shown in generation of cracks due to thermally induced stress states. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.