PAMM · Proc. Appl. Math. Mech. 7, 4030019–4030020 (2007) / DOI 10.1002/pamm.200700620

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

A computational framework of three dimensionalconfigurational-force-driven crack propagation

Ercan Gurses1,∗ and Christian Miehe2,

1 Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA.2 Institut fur Mechanik (Bauwesen), Lehrstuhl I, Universitat Stuttgart, Stuttgart 70550, Germany.

A variational formulation of quasi-static brittle fracture is considered and a new finite-element-based computational frame-work is developed for propagation of cracks in three-dimensional bodies. We outline a consistent thermodynamical frameworkfor crack propagation in elastic solids and show that the crack propagation direction associated with the classical Griffith cri-terion is identified by the material configurational force which maximizes the local dissipation at the crack front. The evolvingcrack discontinuity is realized by the doubling of critical nodes and triangular interface facets of the tetrahedral mesh. Thecrucial step for the success of the procedure is its embedding into an r-adaptive crack-facet reorientation procedure based onconfigurational-force-based indicators in conjunction with crack front constraints. We further propose a staggered algorithmwhich minimizes the stored energy at frozen crack state followed by the successive crack releases at frozen deformation. Thisconstitutes a sequence of positive definite subproblems with successively decreasing overall stiffness, providing a very robustalgorithmic setting in the postcritical range.

1 Variational Formulation of Brittle Crack Propagation

We consider a one-to-one piecewise differentiable transformation Ξt : ΩΓ → BΓ of the reference configuration onto itself.This mapping is considered as the time-dependent parameterization of the medium that accounts for material structuralchanges in the form of a crack propagation. It reflects a time-dependent change of the Lagrangian coordinates θ ∈ ΩΓ toX ∈ BΓ in a sense of a change of material structure. We introduce the material and spatial coordinate maps

Ξt :

{ΩΓ → BΓ

θ �→ X = Ξt(θ)and ξt :

{ΩΓ → SΓ

θ �→ x = ξt(θ)(1)

at time t ∈ R+ and express the deformation map defined in by the composition ϕt(X) = ξt(θ) ◦ Ξ−1t (X) as visualized in

Fig. 1. As a consequence, the deformation gradient F appears as the composition

F = j · J−1 with j = ∇θξt and J = ∇θΞt (2)

of the gradients of the material and spatial coordinate maps introduced in (1). With these definitions at hand, we obtain the

ξt

ϕtΞt

θX

xa

BΓΩΓ

SΓΓt

Γ

Fig. 1 Structural changes and deformation. Both, reference and spatial configurations are independently parameterized bymaterial and spatial maps Ξt and ξt . The change of Ξt in time describes material structural changes.

total time derivative of the above kinematic objects by

ϕ = v − F · V , F = ∇v − F · ∇V , ˙dV = (1 : ∇V)dV (3)

in terms of the spatial and material velocity fields v := ∂

∂tξt and V := ∂

∂tΞt, respectively. These fields govern possible

variations of both the Lagrangian as well as the Eulerian coordinates X ∈ BΓ and x ∈ SΓ.

∗ Corresponding author E-mail: gurses@caltech.edu

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Fig. 2 Prismatic beam subjected to torsion. Geometry, loading, boundary conditions and the deformed mesh.

We focus on an elastic response of the solid with evolving cracks. In order to set up the global constitutive equations forthe crack evolution, we consider a global dissipation analysis in the sense of Coleman’s method. This includes a comparisonof the global power P applied to the solid by external tractions with the global energy storage. We have the global postulate

D := P −d

dtΨ =

∫∂BΓ

t · vdA −d

dt

∫BΓ

ψ(F)dV ≥ 0 , (4)

where ψ denotes the free energy function and t is the traction vector on ∂BΓ. This statement is the demand of the second axiomof thermodynamics in the pure mechanical context. It is the global counterpart to the classical Clausius-Duhem inequalityof continuum thermodynamics. The insertion of the kinematic relationships (3) into (4) with the application of generalizedGauss theorem results in

DivP = 0 in BΓ , DivΣ = 0 in BΓ (5)

and additional traction conditions on the crack surfaces and outer boundaries. These equations cover the equilibrium conditionand the local equation for the Eshelby stress field in the bulk BΓ of the homogeneous elastic solid. Taking into account theconditions (5), we obtain the reduced global dissipation inequality D =

∫∂Γ

g · adS ≥ 0 as an integral over the crack tips.Here, g := lim

|C|→0

∫C

ϕ ·ndS and a are the crack driving force and crack tip velocity, respectively. The crack propagation rate

a needs to be specified by a constitutive assumption. To this end, consider the classical isotropic Griffith-type crack criterionfunction φ(g) = |g| − gc ≤ 0 , where gc is a material parameter specifying the critical energy release per unit length of thecrack. With this notion at hand, an associated evolution equation for the crack evolution may be constructed by introducing anelastic domain for the material forces at the crack tip and a local principle of maximum dissipation. The evolution equationfor the crack propagation reads

a = γ∂gφ(g) = γg

|g|(6)

along with the crack loading-unloading conditions. The algorithmic counterpart of the formulation outlined above requiresthe spatial discretizations of the domain, configurational maps Ξt, ξt and temporal discretization of the evolution equation(6), respectively. In order to obtain a stable setting for this incremental scenario in a typical time interval [tn, tn+1], we applya staggered scheme of energy minimization at frozen crack pattern and a successive crack release by single nodal doubling.We refer to [1, 2] for the details of the algorithm. Here, as an example we consider a torsion test of a concrete prismatic beamwhich has been studied in [3]. The notch is inclined with respect to axes of the beam which yields a double curved cracktrajectory, see Fig. 2. Three dimensional and plan views of the crack trajectories in the undeformed reference configurationare visualized in Fig. 3.

References

[1] C. Miehe and E. Gurses, Int. J. Numer. Meth. Engng. 72, 127–155 (2007).[2] E. Gurses and C. Miehe, submitted to Comput. Methods Appl. Mech. Engrg., (2007).[3] D. R. Brokenshire, PhD Thesis, Cardiff University (1996).

Fig. 3 Prismatic beam subjected to torsion. Visualization of the crack surface in the reference undeformed configuration.

GAMM Sections 4030020