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: