Open Access

The behavior of concrete under high strain rates is often described by plasticity models with softening, which is modeled by a reduction of the yield surface as a function of the local equivalent plastic strain. Among these are the RHT model, the K\&C model and the Johnson-Holmquist concrete model. These models are however local and therefore produce mesh-dependent results. In this contribution, the gradient-enhancement of such models is investigated. First, the mesh-dependency of these local formulations based on the analysis with a modified JH2 model as a representative for these constitutive formulations is demonstrated using a one-dimensional benchmark example. The central difference method is used as solver with a diagonal mass matrix obtained from a Gauß-Lobatto integration. In the benchmark, the width of the damaged zone decreases upon mesh-refinement and the dissipated plastic energy tends to zero. It is further shown that a significantly small safety factor for the critical time step is needed in order to achieve accurate results for the benchmark example. Next, two gradient-enhancement approaches are investigated. The enhancement is based on the inclusion of inertia and damping to the additional Helmholtz equation which enables the use of the central difference method as an explicit solver. In the first formulation, the yield surface and therefore the softening is formulated in terms of a nonlocal equivalent plastic strain. In the second approach, a hardening term which depends on the local equivalent plastic strain is introduced to the modified JH2 model in addition to the nonlocal softening. This approach is inspired by results from gradient plasticity in quasi-static loading scenarios. It is shown that the approach without hardening can still lead to mesh-dependent results while the model that includes hardening successfully inhibits strain localization and leads to a converging dissipated plastic energy. This is further confirmed in a two-dimensional wedge-splitting experiment where the damage pattern produced by the local model is mesh-dependent as well and the dissipated plastic energy tends to zero with mesh-refinement. The proposed nonlocal model with hardening results in a consistent damage pattern and the dissipated plastic energy converges. Furthermore, the nonlocal model with hardening is less sensitive to time step refinement, such that computational efficiency can be improved compared to the local model. The numerical experiments are implemented using the free and open-source tool FEniCSx.

The behavior of concrete under high strain rates is often described by plasticity models with softening, which is modeled by a reduction of the yield surface as a function of the local equivalent plastic strain. Many of these models are local and therefore produce mesh-dependent results. In this contribution, the gradient-enhancement of such models is investigated to mitigate the mesh-dependency. First, the mesh-dependency of these local formulations based on the analysis with a modified JH2 model as a representative for these constitutive formulations is demonstrated using a one-dimensional benchmark example. In the benchmark, the width of the damaged zone decreases upon mesh-refinement and the dissipated plastic energy tends to zero. It is further shown that a significantly small safety factor for the critical time step is needed in order to achieve accurate results for the benchmark example. The first investigated gradient-enhancement approach replaces the equivalent local plastic strain with its nonlocal counterpart. The enhancement is based on the inclusion of inertia and damping to the additional Helmholtz equation which enables the use of the central difference method as an explicit solver. This method successfully distributes the damage over several elements, however, the local equivalent plastic strain still localizes into one cell. The inclusion of hardening with respect to the local equivalent plastic strain inhibits the localization and the dissipated plastic energy converges with mesh-refinement. This is further confirmed in a two-dimensional wedge-splitting experiment and a four-point bending test where the damage pattern produced by the local model is mesh-dependent as well and the dissipated plastic energy tends to zero with mesh-refinement. The proposed nonlocal model with hardening results in a consistent damage pattern and the dissipated plastic energy converges. Furthermore, the nonlocal model with hardening is less sensitive to time step refinement, such that computational efficiency can be improved compared to the local model. The numerical experiments are implemented using the free open source tool FEniCSx and have been made available on Zenodo.

FenicsXConcrete is a Python package for the simulation of mechanical problems. The general PDE solving software FEniCSx is extended with classes describing experimental setups, mechanical problems, thermo-mechanical problems, additive manufacturing and sensors.

The development and integration of user-defined constitutive relationships into finite element (FE) tools using standardized interfaces play a pivotal role in advancing the capabilities of FE solvers for structural mechanics applications. While commercial FE solvers like Abaqus and Ansys have designed their interfaces to provide custom stresses, tangents, and updated history variables, the open-source solver FEniCSx remains efficient only when the constitutive update has an analytical representation. This restricts the application of FEniCSx for non-linear structural mechanics. Since FEniCSx has become a powerful and popular open-source tool for solving partial differential equations, particularly due to its automatic computation of Hessians, we aim to develop a generalized interface to enhance its capability for constitutive modeling. This approach will address complex constitutive equations that require iterative solutions at the quadrature point level. Specific implementation challenges, such as using return-mapping procedures, can then be managed commonly. The provided interface for custom constitutive models offers a versatile way to implement them in various languages, including C++, Python, Rust, and Fortran. Finally, with UMATs for viscoplastic models as an example, we demonstrate how existing user subroutines can be incorporated into the interface and utilized within the FEniCSx framework.

The simulation of concrete under impact and blast loads often relies on local constitutive models, typically formulated as plasticity models that incorporate softening through a scalar damage field. However, these local damage models frequently exhibit mesh-dependent results that fail to converge with mesh refinement. In earlier work, the mesh-dependency of a modified Johnson-Holmquist (JH2) model was effectively mitigated through a gradient-enhanced plasticity formulation in explicit dynamics [1]. The gradient-enhancement method for explicit dynamics, originally introduced in [3], involves modifying Peerlings' additional partial differential equation [2] for nonlocal equivalent plastic strain by incorporating inertia. This modification enables the use of explicit solvers, such as the central difference method. Despite these advancements, the model continues to face challenges, particularly slow convergence rates with mesh refinement, which are difficult to analyze due to the JH2 model's complexity. To address these challenges, this study investigates simpler plasticity models, such as von Mises plasticity and Drucker-Prager plasticity, combined with a nonlocal softening term. By systematically incorporating key characteristics of the JH2 model—namely, pressure-dependent yield surfaces, softening, residual yield strength at full damage and a nonlinear volumetric stress responses via an equation of state (EOS)—this work aims to further refine the gradient-enhanced JH2 model and extend these improvements to more complex plasticity models like the RHT model.