A phase-field model for fractures in nearly incompressible solids

  • Katrin Mang
  • Thomas WickEmail author
  • Winnifried Wollner
Original Paper


Within this work, we develop a phase-field description for simulating fractures in nearly incompressible materials. It is well-known that low-order approximations generally lead to volume-locking behaviors. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor–Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on three numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson’s ratio approximating the incompressible limit, are presented.


Finite elements Phase-field Mixed system Incompressible solids Fracture 

Mathematics Subject Classification

65N30 65N12 35Q74 74R10 



This work has been supported by the German Research Foundation, Priority Program 1748 (DFG SPP 1748) named Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis. Our subproject within the SPP1748 reads Structure Preserving Adaptive Enriched Galerkin Methods for Pressure-Driven 3D Fracture Phase-Field Models (WI 4367/2-1 and WO 1936/5-1). The project number is 392587580.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Applied MathematicsLeibniz Universität HannoverHannoverGermany
  2. 2.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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