FEniCS_Fracture_Kumar_Lopez-Pamies

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FEniCS_Fracture_Kumar_Lopez-Pamies

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FEniCS code for the fracture model of Kumar et al. JMPS 2020: https://www.sciencedirect.com/science/article/pii/S0022509620302623

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# FEniCS_Fracture_Kumar_Lopez-Pamies
! This repository contains an example code in FEniCS to solve a boundary value problem (in parallel) using the new fracture phase-field model introduced in [1]. The example problem studied is the 'Single edge notch tension (SENT)' test in alumina. This phase-field model can describe fracture nucleation in all settings, be it from large pre-existing cracks, small pre-existing cracks,
smooth and non-smooth boundary points, or within the bulk of structures subjected to arbitrary quasistatic loadings, while keeping undisturbed the ability of the standard phase-field formulation to model crack propagation. The model involves the addition of an external configurational force $ce$ to the evolution equation for phase-field variable $z$. This external force depends on the strength surface of the material. In this work, we take the strength surface as the Drucker-Prager surface parametrized by the tensile strength and the compressive strength. Additionally, the force involves a parameter $delta$ that needs to be calibrated to describe fracture nucleation (propagation) from large pre-existing cracks. See Section 4.3.2 in [1] for more details. We have provided an additional code in this repository 'Calibration_of_ce.py' that automatically calibrates the value of the parameter delta. See 'Readme_Calibration.txt' for more details on the calibration.

!**********************************************************************
! Usage:
!
! This code takes as an input of 5 material properties listed below:
! 1. E = Young's modulus
! 2. nu = Poisson's ratio
! 3. Gc = Critical energy release rate
! 4. sts = Tensile strength
! 5. scs = Compressive strength

! Additionally it needs as an input the value of regularization length $eps$ that the user wants to adopt in this boundary value problem. Typically, $eps$ should be chosen so that it much smaller than the average size of the structure. Also, $eps$ should not be chosen to have a larger value than the Irwin's characteristic length (E*Gc/(sts^2)). Otherwise the nucleation (or propagation) from medium-length pre-existing cracks can't be described properly. However, $eps$ can be chosen to be much smaller than the Irwin's characteristic length. Also note, the effective regularization length is even smaller than $eps$ for positive values of $delta$ as defined in equation (21) in [1]. For this chosen value of $eps$, one can determine the value of parameter $delta$ using the code 'Calibration_of_ce.py' included in this repository. The output of that code will also tell the user what mesh size they should use in the regions the fracture will happen in any BVP they wish to solve.

! For this boundary value problem of SENT test, the code will give as output the critical value of stress at which the pre-existing crack propagates (or nucleates or initiates).

!**********************************************************************
! References:

! [1] Kumar, A., Bourdin, B., Francfort, G.A. and Lopez-Pamies, O., 2020. 
      Revisiting nucleation in the phase-field approach to brittle fracture. 
      J. Mech. Phys. Solids, 104027.

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