DOI: 10.1002/pamm.202300237 ISSN: 1617-7061

A phase‐field model for hydrogen‐promoted fracture based on a mixed rate‐type variational setting

Vikas Diddige, Andreas Seupel, Stephan Roth, Bjoern Kiefer
  • Electrical and Electronic Engineering
  • Atomic and Molecular Physics, and Optics


Certain metals experience a substantial deterioration in mechanical properties when exposed to a hydrogen environment, an effect termed hydrogen embrittlement. To understand, predict, and counteract this hydrogen‐assisted material degradation, sufficiently accurate material models are needed. According to the current hypothesis, hydrogen diffusion is driven by gradients of concentration and hydrostatic stress. To capture this, a phase‐field model is formulated as a multi‐field problem coupling deformation, crack propagation, and diffusion to analyze hydrogen‐promoted fracture. Here, the displacements, a fracture‐related phase‐field, the hydrogen lattice occupancy, and the chemical potential are considered as primary field variables. Approaches proposed in the literature often use an extrapolation of the hydrostatic stress calculated at the material point level onto the nodes and later use the B‐matrix to compute the gradient of hydrostatic stress. In order to circumvent this potentially inaccurate extrapolation, the model is recast into a mixed rate‐type variational setting, where the chemical potential—whose gradient governs the hydrogen flux—is obtained from the numerical solution of a saddle point problem. A representative boundary value problem is presented to demonstrate the applicability of the developed numerical framework.

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