4. Algorithmic aspects
4.1 Staggered solution
strategy
Based on the algorithmic paradigm by Miehe et
al 8, the proposed equations solved the weak
formulations of mentioned equations in section 2 using a staggered
strategy(Appendix C)The collection of equations as a result of the
finite element model is nonlinear in order that one has to utilize to
incremental-iterative schemes for calculating the solution. The proposed
model has been realized into application within the software ABAQUS in
order to take advantage of its built-in nonlinear solver using the
Newton–Raphson algorithm along with automatic time-step control
technique.
Respecting to the phase-field model for brittle fracture in
two-dimensional and its extension to ductile one, an 8-node
quadrilateral element is defined with 3 DOF per node, which are
respectively \(u_{x}\), \(u_{y}\) and \(\varphi\). These parameters were
implemented into an interface element in the UEL user subroutine in the
finite element code ABAQUS.
The subroutine is the readout for every element and gets the nodal
values of the element as input. The Abaqus user subroutine employs the
displacement increments to compute incremental strain and called user
subroutine UMAT to achieve the stress increment and the material
jacobian. The latter is needed to matrix A. Implemented formulations of
the stiffness matrix and force vector in section 2 of the element
calculate to matrix A and the vector F and saves them in the arrays
AMATRX and RHS as subroutine’s built-in parameters. In another word,
Abaqus collects contributions from all elements, forms global matrix A
and vector F and finds a correction vector \(\par
\begin{Bmatrix}u\\
\varphi\\
\end{Bmatrix}_{t+t}\)via solving Eq. (39).
4.2 Time integration
To develop accurate, efficient, and stable integration rules for
integrating the set of the constitutive (differential) equations of the
proposed elastoplastic model in section 2, explicit numerical
integration methods were considered. To keep the presentation simple,
the integration strategy has been presented briefly. The generalization
of these integration techniques for a set of PDEs is forthright and
presented afterward.