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We present an exploratory study of the possibilities of the Deep Ritz Method (DRM) for the modeling of strain localization in solids as a sharp discontinuity in the displacement field. For this, we use a regularized strong discontinuity kinematics within a variational setting for elastoplastic solids. The corresponding mathematical model is discretized using Artificial Neural Networks (ANNs). The architecture takes care of the kinematics, while the variational statement of the boundary value problem is taken care of by the loss function. The main idea behind this approach is to solve both the equilibrium problem and the location of the localization band by means of trainable parameters in the ANN. As a proof of concept, we show through both 1D and 2D numerical examples that the computational modeling of strain localization for elastoplastic solids within the framework of DRM is feasible., Comment: The article has 22 pages including 14 figures and 26 references. The manuscript was prepared for submission to Archives of Computational Methods in Engineering

In this paper we tackle the problem of surfactant spreading on a thin liquid film in the framework of isogeometric analysis. We consider a mathematical model that describes this phenomenon as an initial boundary value problem (IBVP) that includes two coupled fourth order partial differential equations (PDEs), one for the film height and one for the surfactant concentration. In order to solve this problem numerically, it is customary to transform it into a mixed problem that includes at most second order PDEs. However, the higher-order continuity of the approximation functions in Isogeometric Analysis (IGA) allows us to deal with the weak form of the fourth order PDEs directly, without the need of resorting to mixed methods. We demonstrate numerically that the IGA solution is able to reproduce results obtained before with mixed approaches. Complex phenomena such as Marangoni-driven fingering instabilities triggered by perturbations are easily captured., Comment: 39 pages, 11 figures

This paper presents a framework for modeling failure in quasi-brittle geomaterials under different loading conditions. A micromechanics-based model is proposed in which the field variables are linked to physical mechanisms at the microcrack level: damage is related to the growth of microcracks, while plasticity is related to the frictional sliding of closed microcracks. Consequently, the hardening/softening functions and parameters entering the free energy follow from the definition of a single degradation function and the elastic material properties. The evolution of opening microcracks in tension leads to brittle behavior and mode I fracture, while the evolution of closed microcracks under frictional sliding in compression/shear leads to ductile behavior and mode II fracture. Frictional sliding is endowed with a non-associative law, a crucial aspect of the model that considers the effect of dilation and allows for realistic material responses with non-vanishing frictional energy dissipation. Despite the non-associative law, a variationally consistent formulation is presented using notions of energy balance and stability, following the energetic formulation for rate-independent systems. The material response of the model is first described, followed by the numerical implementation procedure and several benchmark finite element simulations. The results highlight the ability of the model to describe tensile, shear, and mixed-mode fracture, as well as responses with brittle-to-ductile transition. A key result is that, by virtue of the micromechanical arguments, realistic failure modes can be captured, without resorting to the usual heuristic modifications considered in the phase-field literature. The numerical results are thoroughly discussed with reference to previous numerical studies, experimental evidence, and analytical fracture criteria., Comment: Postprint version. Minor corrections

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topology. Within the computational mechanics community, several studies have treated the issue of modeling fracture using phase fields. Most of these studies have used an approach that implies the lack of a damage threshold. We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach. The formulation is systematically explained within a rigorous variational framework. Subsequently, we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation. Benchmark simulations in two dimensions and three dimensions are then presented. The results show that while an elastic stage and a damage threshold are ensured by the present model, good agreement with the results reported in the literature can be obtained, where such features are generally absent., Comment: 25 pages, 8 figures