Bound-constrained optimization using Lagrange multiplier for a length scale insensitive phase field fracture model

The classical phase field model using the second-order geometric function alpha(phi)=phi(2) (i.e., AT2 model), where phi is an element of[0,1] is an auxiliary phase field variable representing material damage state, has wide applications in static and dynamic scenarios for brittle materials, but nonlinearity and inelasticity are found in its stress-strain curve. The phase field model using the linear geometric function alpha(phi)=phi (i.e., AT1 model), can avoid this, and a linear elastic threshold is available in its stress-strain curve. However, both AT2 and AT1 models are length scale sensitive phase field models, which could have difficulty in adjusting fracture strength and crack band simultaneously through a single parameter (the length scale). In this paper, a generalized quadratic geometric function (linear combination of AT1 and AT2 models) is used in the phase field model, where the extra parameter in this geometric function makes it a length scale insensitive phase field model. Similar to the AT1 model, negative phases can happen in the proposed generalized quadratic geometric function model. To solve this problem, a bound-constrained optimization using the Lagrange multiplier is derived, and the Karush-Kuhn-Tucker (KKT) conditions change from strain energy and maximum history strain energy (an indirect method acting on phase) to phase and Lagrange multiplier (a direct method acting on phase). Several simulations successfully validated the proposed model. A single element analysis and a bar under cyclic loading show the different stress-strain curves obtained from different models. A simulation of Mode I Brazilian test is compared with the experiment conducted by the authors, and two more simulations of Mode II shear test and mixed mode PMMA tensile test are compared with results from the literature.