Introduction

Identification of failure mechanisms and the development of computational methods that precisely estimate complex failure and fracture mechanisms in ductile materials has proven difficult, and many strategies with varying success have been suggested. The phase-field method, also known as the variational approach to fracture, is an approach that has continually been the topic of both scientific interest and paramount importance in engineering applications, which has challenging mathematical and numerical implications.
However, the provision of computational predictive equipment allows for significant financial savings of the cost of experiments, mainly in instances wherein those are extremely complicated, as well as for design optimization.
Following the comprehension review in previous works1, several modeling approaches have been proposed for ductile fracture. For brittle and ductile materials, the basic idea is typically primarily based on the thermodynamic framework first delivered via Griffith.2 The propagation of pre-existing cracks in the phase-field model agrees with the energetic considerations of classical Griffith theory.3,4 The variational approach to brittle fracture, developed by Francfort and Marigo5, to find a solution to the fracture-using minimizing potential energy-based totally on Griffith’s concept of brittle fracture. This method results in Mumford-Shah6. Bourdin et al 7 approved straightforward numerical solutions. An alternative formulation, based on continuum mechanics and thermodynamic theories, become provided by means of Miehe8 and Miehe et al .9
Besides an alternative derivation, Miehe et al 8introduced a crucial mechanism for distinguishing tensile and compressive results on crack growth. The works of Larsen10, Larsen et al 11, Bourdin et al 12, Borden et al 13, and Hofacker and Miehe14demonstrate that this technique can be applied to dynamic fracture and produces results that are consistent with considerable benchmark challenges. Preliminary work to extend the variational approach to ductile materials has been stated in Ambati et al 14,15 and Miehe et al 17,18.They examined the degradation function as a function of the accumulated plastic strain including the elastic modulus, the yield stress, and the strain hardening exponent. The coupled set of stress equilibrium equations and the phase-field evolution are solved at the same time in the work of Miehe and Welschinger9. A staggered scheme is being used in the work of Miehe et al 8 and Aldakheel17. Wherein a local energy history field,\(H\), is adopted as a state variable to guarantee irreversible crack growth.
A related approach is introduced by McAuliffe and Waisman19 where a model that couples the phase-field with the ductile shear band is improved. On this technique, shear bands are formulated the usage of an elastic-perfectly viscoplastic model and fracture is modeled as the degradation of the volumetric elastic stress terms only.
Ductile fracture of elastic-plastic solids turned into an investigation underneath dynamic loading conditions. In the works of Miehe20,21 the point of interest turns out to be placed on reproducing the experimentally determined ductile to brittle failure mode with an increased loading pace. In these works, the whole (free) energy functional is taken because of the accumulation of elastic, plastic and fracture contributions. Recently, Duda et al 22 introduced a phase-field model for quasi-static brittle fracture in elastoplastic solids. T. Gerasimov et al 23 proved that the irreversibility constraint of the crack phase-field is a constrained minimization problem. Bhattacharya et al 24 presented variational gradient damage formulation of ductile failure that naturally couples elasticity, perfect plasticity, and fracture in the rate-independent setting. In this work, small plastic deformation is considered to take place in the location of the notch root or crack tip. Also, in this case, the governing equations in terms of general energy are the sum of elastic, plastic and fracture contributions. The elastic and fracture contributions take the classical form, while the plastic contribution is a delegated function of the accumulated plastic strain.
The objective of this paper is to propose a phase-field formulation of ductile fracture in elastoplastic solids, in the quasistatic boundary problems of linear elastoplasticity with a linear isotropic hardening material. A coupling between the degradation function introduced in15 is investigated. This coupling is shown to play a fundamental role in the correct prediction of some phenomenological aspects of ductile fracture evidenced from available experimental results. Moreover, the model proved to be thermodynamically consistent in 15. One of the significant improvements of the degradation function in this work is \(q\in(0,1]\) parameter which plays a dominating role in the stability of crack propagation.
The development of computer coding via UEL and UMAT subroutines is considered. Analysis of the model yields the definition of an effective fracture strength for one element in the two-dimensional phase-field model. In the second step, the problem of crack initiation and propagation in the one element is extended in the two-dimensional setting. Therefore, based on the findings from the one element case, crack paths and force-displacement curves are derived for the proposed model.

2. Governing Equations

2.1 phase-field summary of brittle fracture of elastic solids:
The phase-field model’s description of brittle fracture drives from the variational formulation of brittle fracture by Francfort and Marigo5, and the regularized formulation of Bourdinet al 7. In Bourdin’s regularized model, the total energy, \(E_{\mathcal{l}}\), of a linear elastic media is: