XXIV ICTAM, 21-26 August 2016, Montreal, Canada
APPROACHES FOR DYNAMIC FRACTURE SIMULATION:
R-ADAPTIVE MATERIAL FORCE AND PHASE-FIELD METHOD
1
Michael Kaliske1 , Christian Steinke1 & Kaan Özenç2
Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
2
ANSYS Inc., Otterfing, Germany
Summary The numerical crack approximation in finite element simulations may be categorized in discrete and
smeared approaches. The discrete approach models a crack as free edges in the meshing with an actual separation
of the elements along the crack surface. The smeared approach represents the crack by a modification of the relation
of stresses and strains in the elements in the vicinity of the crack. This contribution presents the application of state
of the art approaches of each category to dynamic fracture simulations and a detailed comparison of both methods by
evaluation of the simulation results. In respect of the discrete approaches, an r-adaptive node duplication technique
triggered by the evaluation of the “material force“ at the crack tip is introduced. The smeared approach employs
the phase-field method. Both methods are presented and applied to characteristic dynamic fracture examples for
evaluation and comparison of their capabilities.
INTRODUCTION
The first and major step to a comprehensive theory of fracture mechanics was the linking of the formation of a
crack Γ to the dissipation of strain energy by [3]. Furthermore, the knowledge of the material dependent amount
of energy necessary for the formation of a crack allowed the evaluation of strain states of structures in respect of
the propagation of cracks and led to the development of the theory of “material forces“ and their interpretation as
a driving force for crack propagation. With the criterion for crack propagation at hand, the modification of the
spatial discretization led to discrete crack approximation approaches for finite element simulations as described e.g.
in [4].
Figure 1: Continuum Ω with smeared crack approximation Γl and sharp crack Γ
Another approach emerged recently out of the pioneering work of [2]. The formation of cracks, including initiation,
propagation and branching, is embedded into the framework of energy minimization. The additional field parameter
“phase-field“ p(X, t) is introduced and interpreted as the regularized approximation of the crack’s surface Γl . The
relation of the surface energy to the regularized size of the surface of the crack allows the consideration of an
additional dissipative contribution in a general energetic description of the problem. A comprehensive overview on
the existing formulations of the phase-field method is given e.g. in [1].
GOVERNING EQUATIONS
Phase-field method
The main ingredient of the phase-field method is the definition of the part of the strain energy, that is available
for the dissipation into crack surface formation. In this contribution, the spectral decomposition
ψS =
λ
2
2
· hε : 1i+ + µ · ε+ : 1
2
(1)
is employed. Here, the Lamé constants λ and µ, the strain P
tensor ε, the bracket operator hxi+ = x+|x|
and the
2
+
tensile part of the spectrally decomposed strain tensor ε = i hεi i+ ni ⊗ ni are utilized. The irreversibility of the
crack evolution is ensured by the use of a history variable for the crack driving force.
Material force approach
The material forces are evaluated at the nodes of the finite element discretization and present a criterion for the
propagation of the crack. They are evolved from the local form of the simplified material momentum balance
T
∂ψlin 1
T
T
(2)
∇X · Σ = (∇X uT ) b0 − ρ(∇X uT ) ü − ρ(∇X˙uT ) u̇ − u̇ · u̇∇X ρ +
2
∂X exp
in a small strain formulation with the material gradient ∇X (•), the material divergence operator ∇X · (•), the
˙ the displacement vector u, the Eshelby stress tensor Σ, the body forces b0 ,
partial time derivative operator (•),
the density ρ and the free Helmholtz energy density ψlin .
NUMERICAL EXAMPLES
The first numerical example evaluates an experiment on a thin PMMA specimen. In a special setup, the crack
propagation velocity is recorded and can be compared to simulated results.
800
vc [ m
s ]
600
400
Experiment
Material force
Phase-field, coarse mesh
Phase-field, fine mesh
200
0
0
10
20
30
x [mm]
40
50
60
Figure 2: Velocity of crack propagation with respect to the crack length
The second simulation is a purely numerical crack branching benchmark in 2D. A specimen with a pre-existing
crack under two-axial transient loading is studied with both approaches.
(a) Phase-field
(b) Material force
Figure 3: Final path of the crack
CONCLUSION
The presented simulations investigate and compare in depth the characteristics of both approaches in respect of
a realistic approximation of dynamic fracture.
References
[1] Ambati, M., Gerasimov, T., Lorenzis, L.D.: A review on phase-field models of brittle fracture and a new fast hybrid
formulation. Computational Mechanics 55, 383–405 (2015)
[2] Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics
and Physics of Solids 46, 1319–1342 (1998)
[3] Griffith, A.A.: The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London.
Series A 221, 163–198 (1921)
[4] Maugin, G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton
(2010)