TY - JOUR

T1 - A phase-field model for fractures in nearly incompressible solids

AU - Mang, Katrin

AU - Wick, Thomas

AU - Wollner, Winnifried

N1 - Funding Information: 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.

PY - 2020/1

Y1 - 2020/1

N2 - 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.

AB - 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.

KW - Finite elements

KW - Fracture

KW - Incompressible solids

KW - Mixed system

KW - Phase-field

UR - http://www.scopus.com/inward/record.url?scp=85069629136&partnerID=8YFLogxK

U2 - 10.1007/s00466-019-01752-w

DO - 10.1007/s00466-019-01752-w

M3 - Article

AN - SCOPUS:85069629136

VL - 65

SP - 61

EP - 78

JO - Computational mechanics

JF - Computational mechanics

SN - 0178-7675

IS - 1

ER -