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Model-Free Data-Driven Inelasticity
Robert Eggersmann, Trenton Kirchdoerfer, Stefanie Reese, Laurent Stainier, Michael Ortiz
TL;DR
Classical Data-Driven elasticity must be extended to handle material data sets that evolve with history-dependent inelastic behavior. The paper develops memory, differential, and history-variable representations and examines their use across several inelastic material classes. It concludes that many classical viscoelasticity and plasticity models can be represented with differential and/or history-variable data sets, while history matching is practical mainly for short prior histories.
Problem
Data-Driven inelasticity must represent evolving, history-dependent material data sets while remaining rigorous and practical.
Method
The paper extends Data-Driven elasticity by conditioning material data on full deformation histories, short stress-strain histories, history variables, or combinations of these paradigms.
Results
Many classical viscoelasticity and plasticity models can be represented using differential and/or history-variable material data sets.
Takeaways & Limitations
History repositories permit offline sampling but increase data dimensionality, making history matching practical mainly for short prior histories.
Takeaways & Limitations
History variables encode only partial historical information and need not have a specific physical meaning, reflecting a modeling trade-off.
Abstract
from arXiv · showhide
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.
1. Introduction
Data-Driven inelasticity extends direct material-data formulations to materials whose response depends on deformation history. The paper frames evolving-data representation as a balance between rigor, practicality, and coverage of inelastic behavior.
- Data-Driven problems minimize distance between material data and compatible, equilibrated strain-stress fields, recovering classical linear-elastic solutions.
- Unlike conventional constitutive approaches, the formulation uses material data directly without modeling, reducing, or otherwise manipulating the data.
- Materials with memory provide a general history-based representation, while internal-variable approaches encode effects of history through the material element’s current microstructure.
- Inelastic material data sets evolve over time because attainable stress-strain pairs depend on the material’s prior stress and strain history.
- The paper investigates memory, differential, and history-variable representations, including combinations, for viscoelasticity, viscoplasticity, and plasticity.
- The resulting problems minimize phase-space distance between evolving data sets and time-dependent constraint sets, with attention to numerical implementation and convergence.
2. Background: Data-Driven elasticity
Data-Driven elasticity formulates mechanics directly in phase space using material data and universal compatibility and equilibrium constraints. A projection-based iteration alternates between material-data association and constraint enforcement.
- The internal state is represented by local stress-strain pairs in local phase spaces, assembled into a global phase space.
- Compatibility and equilibrium constraints are material-independent and define the subspace of compatible and equilibrated internal states.
- Instead of inferring a constitutive law from experimental material points, Data-Driven mechanics formulates boundary-value problems directly from the material data.
- The Data-Driven problem finds the compatible and equilibrated state closest to the global material data set, equivalently minimizing phase-space distance.
- Arbitrary phase-space data sets, including graphs, point sets, fat sets, and ranges, are permitted; constitutive graphs recover classical mechanics as a special case.
- The fixed-point solver alternates projection onto the material data set and the constraint set until the associated data points no longer change.
3. Extension to inelasticity
Data-Driven inelasticity makes both the constraint set and material data set time-dependent, with attainable stress–strain pairs conditioned by prior material history. The paper develops memory, internal/history-variable, and differential representations to characterize this evolving data practically.
- Time-dependent Data-Driven formulation: At each discrete time, the inelastic Data-Driven problem seeks compatible and equilibrated states closest to the time-dependent material data set.The constraint set changes with applied loads, while attainable local stress–strain pairs change with material history.
- Materials with memory: Materials with memory represent attainable stress–strain pairs by conditioning them on the entire prior strain history.Fading memory can permit truncation beyond a decay time, but tracking and sampling long histories may remain onerous.
- Internal variables: Internal-variable representations reduce history parametrization by conditioning the material data set on the prior internal state and governing its evolution through kinetic relations.The resulting equations determine the current internal variables and stress for a given strain and previous internal state.
- History variables: History variables retain the efficiency of internal-state parametrization without requiring the variables to have a specific physical meaning.Their role is to record partial information about stress and strain histories, making their selection an approximation-theory question rather than a material-modeling assumption.
- Differential representations: Differential representations condition material data sets on short stress and strain histories, potentially requiring smaller parametrizations than representations based only on long strain histories.When constitutive relations are sufficiently differentiable and behavior is stable, differential and internal-variable representations are equivalent; differential models need only stress and strain data.
4. Numerical examples: Viscoelasticity
The viscoelastic examples use differential material-data representations, showing exact representation for the Standard Linear Solid and convergence toward its reference response in a truss.
- Differential representation: Viscoelasticity's smooth kinetic equations and stable equilibrium manifold make its evolving data sets suitable for differential representations.The paper identifies differential representations as particularly appropriate for viscoelastic solids.
- Standard Linear Solid: First-order differential representations exactly represent the Standard Linear Solid, while generally approximating unknown material behavior.For fixed strain and stress, the representation defines a linear subspace of phase space with dimension Rme.
- Relaxation test: In the relaxation test, the constraint set fixes strain while successive material data sets translate downward in phase space, tracing the bar's relaxation curve.The initial data set gives the instantaneous response, and later intersections with the constraint set produce the relaxation response.
- Truss convergence: The truss example contains 1,246 bars with loads ramped, held, reversed, and removed over a 100-time-unit simulation.Standard Linear Solid data are randomized on the fly to generate the material data sets.
- Truss convergence: Increasing the number of material data points makes viscoelastic truss time histories converge toward the reference Standard Linear Solid solution.The comparison includes output-node displacements and reaction-force histories.
- Truss convergence: 50 independent randomizations yield a quadratic convergence rate, twice the linear rate characteristic of elastic problems.Convergence is assessed using a weighted ℓ2 error over the time steps.
5. Numerical examples: Plasticity
The plasticity example shows that isotropic-kinematic linear hardening requires a mixed differential-hereditary material-data representation, with a history variable capturing additional state information. Data-Driven solutions for a plastic truss converge toward the reference hardening solution as material-data resolution increases, with roughly linear convergence.
- Representational requirements: Plasticity is not amenable to a strict differential representation and requires history variables in addition.
- Material model: The isotropic-kinematic linear-hardening solid uses an internal inelastic strain, accumulated plastic strain, stored energy of cold work, and elastic moduli in its material description.
- Material model: The discrete plasticity formulation imposes Kuhn-Tucker yielding and loading-unloading conditions within a fully implicit maximum-dissipation problem.
- Representational requirements: The attainable data set at time t_k+1 is characterized by the previous stress-strain pair and history variable q_e,k, combining differential and history-variable representations.
- Representational requirements: For fixed previous state variables, the mixed representation defines a linear phase-space subspace and exactly represents the isotropic-kinematic plastic solid, whereas general plastic solids may only be approximated.
- Convergence analysis: The plastic-truss time histories converge toward the reference isotropic-kinematic hardening solution as the number of material data points increases, with a roughly linear computed convergence rate.The convergence statistics compile 50 independent runs.
6. Summary and concluding remarks
The paper extends Data-Driven mechanics to history-dependent inelasticity, where material data sets evolve with time, and evaluates representations for capturing that evolution. It also outlines extensions involving history matching, goal-oriented data acquisition, fast search, and data fidelity.
- Contributions: The evolving data set is the central distinction between Data-Driven elasticity and inelasticity because material response depends on deformation history.The extension addresses irreversible, history-dependent materials rather than fixed elastic data sets.
- Contributions: Three paradigms condition material data on deformation memory, short stress–strain histories, or history variables encoding partial historical information.Combinations of these paradigms are also considered for representing inelastic materials.
- Contributions: Many classical viscoelastic and plasticity models can be represented with differential and/or history-variable data sets, but these representations approximate complex material behavior.The paper frames increasing historical information as the basis for improving representation accuracy, without providing a rigorous analysis.
- History matching: History matching compares short material histories directly and assigns confidence weights according to their distance from the actual prior history.For a Standard Linear Solid, two-time stress–strain histories form the local history repository; prior histories can be sampled offline.
- Extensions: Goal-oriented self-consistent acquisition generates data covering the phase-space region relevant to a specific problem while solving for the corresponding Data-Driven solution.The approach uses non-homogeneous strain fields, including fields measured through Digital Image Correlation.