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.