On the implementation of finite deformation gradient-enhanced damage models


Classic works on the modelling of damage mainly consider damage to be a local effect where an effective area reduction of the stress-bearing region leads to material degradation. From a modelling perspective, a significant drawback of classic continuum damage formulations is their inherently local character. In fact, in the context of the finite element method, the purely local consideration of continuum damage leads to mesh-dependency and hence to mostly meaningless results. Therefore, in order to regularize the localized damage zone, different approaches have been proposed. For modern applications, non-local continuum theories have been developed. Overall, non-locality can generally be incorporated by either integral or gradient-type extensions of the underlying continuum formulation. Gradient-type formulations provide several advantages over integral-type non-local formulations, thus making them a more popular choice for damage regularization. Recent applications and extensions of the particular damage formulation are mainly based on the implementation of an elaborate user element formulation within commercial finite element codes such as Abaqus. Even so, the use of a user element formulation—enabled in Abaqus via the UEL subroutine—has several drawbacks.

To address this, researchers from the Institute of Mechanics at TU Dortmund University in Germany, Dr. Richard Ostwald and Professor Andreas Menzel, together with Professor Ellen Kuhl at the Department of Mechanical Engineering at Stanford University, proposed a comprehensive framework for the efficient implementation of finite deformation gradient-regularized damage formulations in existing finite element codes. The researchers exemplarily focused on the finite element software Abaqus for the implementation, where they utilized the two most general Abaqus subroutines for this particular coupled problem, i.e. the UMAT and UMATHT subroutine. Their work is currently published in the research journal Computational Mechanics.

Technically, their framework was based on the observation that the underlying damage-related balance equation is a partial differential equation of elliptic type, as is the steady-state heat equation. Consequently, the researchers proceeded to show that the damage regularization could be accomplished by using the heat-equation solution capabilities included in thermomechanically coupled finite element formulations.

The research team reported that the numerical solution framework they presented allowed for the implementation of gradient enhanced damage formulations within commercial finite element software without the need for user element subroutines, thereby eliminating all of the disadvantages associated with them. Moreover, the researchers were able to demonstrate a boundary value problem reflecting an indentation test to emphasize that the framework allowed them to directly combine regularized damage with advanced element features such as contact.

In summary, the study introduced a comprehensive framework for the user element-free implementation of gradient-enhanced damage formulations in existing finite element codes with an exemplary implementation in the finite element tool Abaqus. Most important, the introduced framework circumvents the cumbersome user element formulations and their associated drawbacks. In an interview with Advances in Engineering, Dr. Richard Ostwald, the corresponding author, highlighted that, by considering the additionally provided algorithms for the automatic numerical computation of the required Abaqus UMAT tangent contributions for general thermomechanically coupled user materials, their work allowed for a quick and efficient implementation of any local constitutive model that undergoes regularized damage, at the same time allowing for a direct combination with advanced element features such as contact or incompressibility, or even structural elements or remeshing schemes.

On the implementation of finite deformation gradient-enhanced damage models - Advances in Engineering
Figure 1: Asymmetrically notched plate undergoing damage under tension. Even though the load-displacement curve reveals severe material softening in state C (left), a mesh-objective distribution of the damage variable d is obtained by exploiting a thermo-mechanically coupled finite element formulation for the gradient-based regularisation of damage, shown using a fine mesh (mid) and coarse mesh (right).
On the implementation of finite deformation gradient-enhanced damage models - Advances in Engineering
Figure 2: Tensile test combining regularised damage with a mixed finite element formulation, suitable for the mesh-objective simulation of damage in incompressible materials. Stress field (left), non-local damage variable computed using the heat equation (mid), and independent pressure field (right).

About the author

Richard Ostwald completed his PhD thesis at the Institute of Mechanics, TU Dortmund University, Germany, under the supervision of Professor Andreas Menzel in 2015. He continued his postdoctoral career with a research stay at Stanford University, CA, with Professor Ellen Kuhl and currently serves as interim professor and interim head of the Department of Engineering Mechanics at Bremen University, Germany. He has published numerous articles, textbook contributions, and is managing editor of the international journal Surveys for Applied Mathematics and Mechanics, published by the International Association of Applied Mathematics and Mechanics.

His contributions to the field of computational mechanics, ranging from micro-scale models for complex materials to advanced process simulations, have been honored by, among others, the German Association of Computational Mechanics (GACM) and Polish Association of Computational Mechanics (PACM). You can find Richard Ostwald’s website at www.richardostwald.de


Richard Ostwald, Ellen Kuhl, Andreas Menzel. On the implementation of finite deformation gradient-enhanced damage models. Computational Mechanics (2019) volume 64: page 847–877.

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