Model-Free Data-Driven Inelasticity (1808.10859v2)

Abstract: We extend the Data-Driven formulation of problems in elasticity of Kirchdoerfer and Ortiz (2016) to inelasticity. This extension differs fundamentally from Data-Driven problems in elasticity in that the material data set evolves in time as a consequence of the history dependence of the material. We investigate three representational paradigms for the evolving material data sets: i) materials with memory, i.e., conditioning the material data set to the past history of deformation; ii) differential materials, i.e., conditioning the material data set to short histories of stress and strain; and iii) history variables, i.e., conditioning the material data set to ad hoc variables encoding partial information about the history of stress and strain. We also consider combinations of the three paradigms thereof and investigate their ability to represent the evolving data sets of different classes of inelastic materials, including viscoelasticity, viscoplasticity and plasticity. We present selected numerical examples that demonstrate the range and scope of Data-Driven inelasticity and the numerical performance of implementations thereof.

• The paper introduces a novel data-driven framework that models inelasticity without relying on traditional empirical constitutive equations.
• It evaluates three paradigms—materials with memory, differential materials, and history variables—to effectively capture path-dependent behavior.
• Numerical results demonstrate rapid convergence and accurate simulation of viscoelastic and plastic behaviors, highlighting its potential for large-scale applications.

Overview of "Model-Free Data-Driven Inelasticity"

The paper "Model-Free Data-Driven Inelasticity" by Eggersmann et al. extends the Data-Driven approach previously developed for elastic materials to include inelastic behavior characterized by history dependence. This work offers a novel perspective that diverges from traditional constitutive models, which typically rely on empirical formulations to describe material behavior. Instead, this Data-Driven methodology operates directly from experimental data, featuring evolving material data sets to accommodate the complex nature of inelasticity.

Key Concepts and Methodologies

The cornerstone of the approach is the extension of the Data-Driven framework to address inelastic materials. This is achieved by configuring material data sets that evolve over time due to the path-dependent nature of inelasticity. The paper evaluates three specific paradigms for this purpose:

1. Materials with Memory: This paradigm considers the entire deformation history, portraying a classical perspective of non-linear viscoelasticity using hereditary integrals. The approach may become computationally burdensome due to the need to track long histories, particularly when implementing fading memory laws.
2. Differential Materials: In this approach, the material response is defined by differential equations linking stress and strain over short history intervals. This reduces computational demand and could potentially offer a more tractable way of modeling complex inelastic behavior.
3. History Variables: This paradigm finds utility in transforming internal variables into heuristic history variables. These variables are not inherently tied to physical processes but are selected to capture the essence of material response history in a reduced manner. This approach serves to effectively approximate material behavior where full historical tracking is impractical.

Each paradigm can be utilized independently or in combination to adapt to different classes of inelastic materials like viscoelasticity, viscoplasticity, and plasticity, making the framework versatile and broad in application.

Numerical Implementation and Results

The authors demonstrated the feasibility and applicability of their approach through several numerical examples:

• Viscoelasticity: Utilizing the differential paradigm, the standard linear solid (SLS) model was simulated. This exercise illustrated that first-order differential representations could precisely capture viscoelastic behavior, demonstrating the methodology's strength in handling such systems efficiently.
• Plasticity: The paper tackles isotropic-kinematic hardening models as exemplars of plastic behavior. For these scenarios, a combination of differential representation and history variables were shown to be necessary to accurately represent the material states, especially under non-linear hardening conditions.

Robust numerical results on problems involving complex truss structures highlighted the ability of this new Data-Driven approach to converge rapidly to solutions, exhibiting behavior comparable to classical models while bypassing the need for explicit constitutive equations.

Implications and Future Directions

The research opens new avenues in material modeling for inelastic systems by emphasizing data-centric approaches. By reducing reliance on traditional empirical models, the method can adapt to a wider range of materials, including those exhibiting complex or poorly understood behaviors.

Going forward, incorporating aspects of machine learning—perhaps through advanced search algorithms or other AI techniques—can enhance the framework's efficiency. Additionally, addressing multi-fidelity data structures and history-matching methods may offer further advancements in accounting for varying data quality and history effects.

The paper prompts intriguing challenges, notably concerning the scalability of data-driven solutions to large-scale systems and the systematic handling of uncertainty in data acquisition, necessary for practical implementation in scientific and industrial applications. These ongoing developments mark a progressive step towards integrating empirical data into computational mechanics, hoping to influence future theoretical and computational advancements in modeling inelastic phenomena.