Learning viscoplastic constitutive behavior from experiments: II. Dynamic indentation

Aakila Rajan; Andrew Akerson; Daniel Casem; Kaushik Bhattacharya

arxiv: 2510.27570 · v2 · submitted 2025-10-31 · ❄️ cond-mat.mtrl-sci

Learning viscoplastic constitutive behavior from experiments: II. Dynamic indentation

Andrew Akerson , Aakila Rajan , Daniel Casem , Kaushik Bhattacharya This is my paper

Pith reviewed 2026-05-18 03:03 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords viscoplastic constitutive behaviordynamic indentationinverse problembalance lawsadjoint methodcontact constraintmaterials characterizationsteel and aluminum alloys

0 comments

The pith

A constrained inverse problem method identifies viscoplastic constitutive behavior from dynamic indentation experiments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends a prior method for inferring material constitutive relations from full-field observations to dynamic indentation experiments that involve contact. The task is posed as an optimization problem that seeks the constitutive model minimizing mismatch with observed data while strictly enforcing the balance laws of mechanics. The forward problem is formulated as a boundary value problem for the indentation setup, and the adjoint method supplies the sensitivities needed by the optimizer. Contact is treated as a nonholonomic constraint through introduction of a Lagrange multiplier together with a slack variable. The procedure is first verified on synthetic data and then applied to experimental records for rolled homogeneous armor steel and a polycrystalline aluminum alloy.

Core claim

The constitutive behavior of viscoplastic materials is recovered by solving an indirect inverse problem constrained by the balance laws, where the forward problem is a boundary value problem corresponding to the dynamic indentation experiment, sensitivities are obtained via the adjoint method, and contact is incorporated through a Lagrange multiplier and slack variable.

What carries the argument

Adjoint-method computation of objective sensitivities with respect to the constitutive representation inside a dynamic boundary-value problem that includes nonholonomic contact constraints.

If this is right

The method recovers constitutive models from full-field experimental data without presupposing a specific functional form.
Balance laws are enforced exactly throughout the identification process.
Contact conditions are handled rigorously, extending the approach to surface-loading experiments.
The procedure is demonstrated on both synthetic data and real measurements from armor steel and aluminum alloy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be applied to other high-rate contact problems such as ballistic impact or machining.
Recovered models may improve finite-element predictions of deformation and failure in structural components.
Integration with additional full-field diagnostics like digital image correlation at higher frame rates would further constrain the inverse solution.

Load-bearing premise

The chosen parameterization of the constitutive behavior is flexible enough to capture the true material response.

What would settle it

Independent uniaxial or other standardized tests performed on the same steel and aluminum specimens, compared directly against the stress-strain response predicted by the recovered model.

Figures

Figures reproduced from arXiv: 2510.27570 by Aakila Rajan, Andrew Akerson, Daniel Casem, Kaushik Bhattacharya.

**Figure 2.** Figure 2: Demonstration with synthetic data for the the data generated with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Response to an independent uniaxial tensile test using the synthetic parameters [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Demonstration on linearized synthetic data generated from [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Results for experiments with RHA steel. (a ) Indentation velocity vs time for the two [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Characterization results from experiments on aluminium alloy Al 6061-T6. (a) Inden [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Raw and filtered indentation force F, Fe vs indentation depth δ for a typical dynamic indentation simulation. This completes the update to the displacement and velocity fields for the timestep. We then update the plasticity variables through an implicit backwards Euler discretization. For this, we employ a predictor-corrector scheme [25] to solve point-wise at each quadrature point, 0 ∈ σM(ε n+1|xg , εp,n+… view at source ↗

read the original abstract

We continue the development of a method to accurately and efficiently identify the constitutive behavior of complex materials through full-field observations that we started in Akerson, Rajan and Bhattacharya (2024). We formulate the problem of inferring constitutive relations from experiments as an indirect inverse problem that is constrained by the balance laws. Specifically, we seek to find a constitutive behavior that minimizes the difference between the experimental observation and the corresponding quantities computed with the model, while enforcing the balance laws. We formulate the forward problem as a boundary value problem corresponding to the experiment, and compute the sensitivity of the objective with respect to the model using the adjoint method. In this paper, we extend the approach to include contact and study dynamic indentation. Contact is a nonholonomic constraint, and we introduce a Lagrange multiplier and a slack variable to address it. We demonstrate the method on synthetic data before applying it to experimental observations on rolled homogeneous armor steel and a polycrystalline aluminum alloy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends their adjoint inverse method with a contact treatment and applies it to real dynamic indentation data, but the synthetic tests likely stay inside the model space so real-material recovery remains a fit rather than a verified law.

read the letter

This paper takes the adjoint-constrained inverse approach from their part-I work and adds a Lagrange multiplier plus slack variable to handle nonholonomic contact. That lets them set up the forward problem as a dynamic indentation boundary-value problem and recover the viscoplastic response by minimizing data misfit while enforcing balance laws through the adjoint sensitivity. They first check it on synthetic data, then run it on full-field experimental measurements from rolled homogeneous armor steel and a polycrystalline aluminum alloy. The contact extension is the concrete addition, and showing the pipeline on actual high-rate experiments is the practical step forward. It gives a route to rate-dependent models without running many separate calibration tests, which matters for impact modeling. The formulation stays consistent with the balance laws, and the abstract reports successful recovery in both cases. The main limitation is the one the stress-test note flags. If the synthetic constitutive laws are generated inside the same function space the optimizer searches, then successful recovery only proves internal consistency. For the real alloys we still do not know whether the true viscoplastic behavior sits outside that space; if it does, the recovered model is the best projection within the chosen representation rather than the physically correct law. The paper would be tighter if it spelled out the exact parameterization and included at least one out-of-space synthetic case. This is aimed at people who build constitutive models for high-strain-rate problems in materials and structural mechanics. A reader already working on inverse methods or full-field calibration will find the contact handling and the experimental application useful. The work is grounded enough and the application relevant enough that it deserves a serious referee, even if reviewers will press on regularization choices, quantitative error metrics, and the representation limits.

Referee Report

2 major / 1 minor

Summary. The manuscript extends the authors' prior adjoint-constrained inverse-problem framework for inferring viscoplastic constitutive relations from full-field observations. Contact in dynamic indentation is treated as a nonholonomic constraint by introducing a Lagrange multiplier and slack variable. The forward problem is solved as a boundary-value problem, sensitivities are obtained via the adjoint method, and the constitutive behavior is recovered by minimizing the data misfit while enforcing balance laws. The approach is first validated on synthetic data and then applied to experimental dynamic indentation results for rolled homogeneous armor steel and a polycrystalline aluminum alloy.

Significance. If the recovered models satisfy the contact constraint to high precision and reproduce the observed fields while obeying balance laws, the work would provide a practical, physics-constrained route to data-driven constitutive identification under dynamic contact conditions. The adjoint formulation and the handling of the nonholonomic constraint are technically attractive features that could reduce reliance on assumed functional forms in high-strain-rate materials characterization.

major comments (2)

[Formulation of the inverse problem] Formulation of the inverse problem (abstract and the paragraph describing the indirect inverse problem): the central claim that the method recovers the constitutive behavior of the experimental materials requires that the chosen parameterization (basis or network) be rich enough to represent the true viscoplastic response. The manuscript should explicitly state the function space searched by the optimizer and either demonstrate that the true response lies inside it or discuss the consequences if it does not; otherwise the experimental results on steel and aluminum are best interpreted as projections onto that space rather than the physically correct law.
[Synthetic data demonstration] Synthetic data demonstration section: the paper must clarify whether the constitutive laws used to generate the synthetic data belong to the same function space as the parameterization employed in the inverse problem. If the synthetic cases are generated in-space, successful recovery only verifies internal consistency and does not address the risk of biased recovery when the true response lies outside the search space, which directly affects in the experimental applications.

minor comments (1)

[Abstract] The abstract would benefit from a concise statement of the specific parameterization adopted for the viscoplastic relation and the principal quantitative findings on the two experimental materials.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate planned revisions to clarify the function space and its implications.

read point-by-point responses

Referee: Formulation of the inverse problem (abstract and the paragraph describing the indirect inverse problem): the central claim that the method recovers the constitutive behavior of the experimental materials requires that the chosen parameterization (basis or network) be rich enough to represent the true viscoplastic response. The manuscript should explicitly state the function space searched by the optimizer and either demonstrate that the true response lies inside it or discuss the consequences if it does not; otherwise the experimental results on steel and aluminum are best interpreted as projections onto that space rather than the physically correct law.

Authors: We agree that the function space must be stated explicitly for proper interpretation. In the revision we will add a subsection detailing the specific parameterization (basis functions or network architecture and hyperparameters) used for the viscoplastic response. We will also add discussion clarifying that recovered models are the best fit within this space subject to the balance laws and contact constraints, and note the consequences if the true response lies outside it. revision: yes
Referee: Synthetic data demonstration section: the paper must clarify whether the constitutive laws used to generate the synthetic data belong to the same function space as the parameterization employed in the inverse problem. If the synthetic cases are generated in-space, successful recovery only verifies internal consistency and does not address the risk of biased recovery when the true response lies outside the search space, which directly affects in the experimental applications.

Authors: We acknowledge the distinction. The synthetic cases were generated from the same parameterized family to verify recovery under known conditions. The revision will explicitly state this and add discussion of the resulting limitations for experimental data, where the recovered law is the optimal projection onto the chosen space. We will note that tests with out-of-space synthetics remain future work. revision: yes

Circularity Check

0 steps flagged

Inverse problem formulation is self-contained optimization with no reduction of results to inputs by construction

full rationale

The paper formulates inferring constitutive relations as an indirect inverse problem minimizing data misfit subject to balance laws, solved via adjoint sensitivity, with an extension to handle contact via Lagrange multiplier and slack variable. Synthetic demonstrations verify the numerical procedure on data generated from a chosen law, which is standard method validation rather than a circular recovery. Real-material application to indentation experiments on steel and aluminum produces the best-fit behavior within the chosen parameterization. The self-citation to the 2024 prior work introduces the base approach but is not load-bearing for the new contact treatment or results here; the central claim remains the output of the stated optimization and does not reduce to a fitted input renamed as prediction or to a self-citation chain. No self-definitional, ansatz-smuggling, or renaming steps are present in the abstract or described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the assumption that the balance laws are known exactly and that the experimental fields are measured with sufficient spatial and temporal resolution to constrain the constitutive response; no new physical entities are postulated and the only free parameters are those internal to the chosen constitutive representation, which the inverse procedure is designed to determine.

axioms (2)

domain assumption The balance laws of continuum mechanics hold pointwise throughout the specimen during the experiment.
Invoked when the inverse problem is stated as a constrained optimization that enforces the balance laws (abstract).
domain assumption The contact condition can be enforced exactly by introducing a Lagrange multiplier and slack variable without introducing additional modeling error.
Stated when the authors extend the formulation to include contact (abstract).

pith-pipeline@v0.9.0 · 5702 in / 1590 out tokens · 34869 ms · 2026-05-18T03:03:28.838469+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We formulate the problem of inferring constitutive relations from experiments as an indirect inverse problem that is constrained by the balance laws... We demonstrate the method on synthetic data before applying it to experimental observations on rolled homogeneous armor steel and a polycrystalline aluminum alloy.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Contact is a nonholonomic constraint, and we introduce a Lagrange multiplier and a slack variable to address it.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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B. Si, Z. Li, Y. Yang, L. Qiao, E. Liu, G. Xiao, and X. Shu. Dynamic indentation testing and characterization of metals based on the split Hopkinson pressure bar (SHPB) device. Mechanics of Materials, 177:104550, 2023

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J. Simo and T. Laursen. An augmented lagrangian treatment of contact problems involving friction.Computers & Structures, 42(1):97–116, 1992

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Wriggers, J

P. Wriggers, J. Schr¨ oder, and A. Schwarz. A finite element method for contact using a third medium.Computational Mechanics, 52, 10 2013

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C. Xiao, Y. Deng, and G. Wang. Deep-learning-based adjoint state method: Method- ology and preliminary application to inverse modeling.Water Resources Research, 57(2):e2020WR027400, 2021. A Numerical method for the governing equations We detail the finite element discretization and numerical scheme we use to solve the forward prob- lem. We consider a quad...

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[1] [1]

A. Akerson. Optimal structures for failure resistance under impact.Journal of the Mechanics and Physics of Solids, 172:105172, 3 2023

work page 2023

[2] [2]

Akerson and T

A. Akerson and T. Liu. Mechanics, modeling, and shape optimization of electrostatic zipper actuators.Journal of the Mechanics and Physics of Solids, 181:105446, 10 2023

work page 2023

[3] [3]

Akerson, A

A. Akerson, A. Rajan, and K. Bhattacharya. Learning constitutive relations from experiments:

work page

[4] [4]

pde constrained optimization.Journal of the Mechanics and Physics of Solids, 201:106128, 2025

work page 2025

[5] [5]

R. J. Anton and G. Subhash. Dynamic Vickers indentation of brittle materials.Wear, 239:27– 35, 2000

work page 2000

[6] [6]

Arndt, W

D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, and D. Wells. The deal.ii finite element library: Design, features, and insights.Computers & Mathematics with Applications, 81:407–422, 2021. 14

work page 2021

[7] [7]

Bhattacharya, B

K. Bhattacharya, B. Liu, A. Stuart, and M. Trautner. Learning markovian homogenized models in viscoelasticity.Multiscale Modeling & Simulation, 21(2):641–679, 2023

work page 2023

[8] [8]

Camacho and M

G. Camacho and M. Ortiz. Adaptive lagrangian modelling of ballistic penetration of metallic targets.Computer Methods in Applied Mechanics and Engineering, 142(3):269–301, 1997

work page 1997

[9] [9]

Casem and E

D. Casem and E. Retzlaff. A kolsky bar for high-rate indentation.Journal of Dynamic Behavior of Materials, 9(3):300–314, 2023

work page 2023

[10] [10]

Clayton, J

J. Clayton, J. Lloyd, and D. Casem. Simulation and dimensional analysis of instrumented dynamic spherical indentation of ductile metals.International Journal of Mechanical Sciences, 251:108333, 2023

work page 2023

[11] [11]

Y. Deng, S. Ren, K. Fan, J. M. Malof, and W. J. Padilla. Neural-adjoint method for the inverse design of all-dielectric metasurfaces.Optics Express, 29(5):7526–7534, 2021

work page 2021

[12] [12]

G. Gray, S. R. Chen, W. Wright, and M. Lopez. Constitutive equations for annealed metals under compression at high strain rates and high temperatures. Technical report, Los Alamos National Laboratory Report: LA-12669-MS, 1994

work page 1994

[13] [13]

H. Hertz. Ueber die ber¨ uhrung fester elastischer k¨ orper.Journal f¨ ur die reine und angewandte Mathematik, 1882(92):156–171, 1882

work page

[14] [14]

H¨ ueber, G

S. H¨ ueber, G. Stadler, and B. I. Wohlmuth. A primal-dual active set algorithm for three- dimensional contact problems with coulomb friction.SIAM Journal on Scientific Computing, 30(2):572–596, 2008

work page 2008

[15] [15]

K. L. Johnson.Contact Mechanics. Cambridge University Press, 1985

work page 1985

[16] [16]

Kikuchi and J

N. Kikuchi and J. T. Oden.Contact Problems in Elasticity. Society for Industrial and Applied Mathematics, 1988

work page 1988

[17] [17]

B. J. Koeppel and G. Subhash. An experimental technique to investigate the dynamic inden- tation hardness of materials.Experimental Techniques, 21:16–18, May 1997

work page 1997

[18] [18]

Kovachki, Z

N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anand- kumar. Neural operator: Learning maps between function spaces with applications to pdes. Journal of Machine Learning Research, 24(89):1–97, 2023

work page 2023

[19] [19]

Laursen.Computational Contact and Impact Mechanics: Fundamentals of Modeling Inter- facial Phenomena in Nonlinear Finite Element Analysis

T. Laursen.Computational Contact and Impact Mechanics: Fundamentals of Modeling Inter- facial Phenomena in Nonlinear Finite Element Analysis. Engineering online library. Springer Berlin Heidelberg, 2003

work page 2003

[20] [20]

Lee and K

A. Lee and K. Komvopoulos. Dynamic spherical indentation of elastic-plastic solids.Interna- tional Journal of Solids and Structures, 146:180–191, 2018

work page 2018

[21] [21]

B. Liu, E. Ocegueda, M. Trautner, A. M. Stuart, and K. Bhattacharya. Learning macroscopic internal variables and history dependence from microscopic models.Journal of the Mechanics and Physics of Solids, 178:105329, 2023

work page 2023

[22] [22]

Lubliner.Plasticity Theory

J. Lubliner.Plasticity Theory. Dover books on engineering. Dover Publications, 2008. 15

work page 2008

[23] [23]

Meyer and D

H. Meyer and D. S. Kleponis. An analysis of parameters for the johnson-cook strength model for 2-in-thick rolled homogeneous armor.Army Research Laboratory, Adelphi, MD, Report No. ARL-TR-2528, 2001

work page 2001

[24] [24]

Molinari, M

J.-F. Molinari, M. Ortiz, R. Radovitzky, and E. Repetto. Finite-element modeling of dry sliding wear in metals.Engineering Computations (Swansea, Wales), 18, 10 2002

work page 2002

[25] [25]

M. Nilsson. Dynamic hardness testing using a split hopkinson pressure bar apparatus, 2002. Swedish Defense Research Agency: SE-147 25

work page 2002

[26] [26]

Ortiz and L

M. Ortiz and L. Stainier. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171(3-4):419–444, 1999

work page 1999

[27] [27]

S. Ren, W. Padilla, and J. Malof. Benchmarking deep inverse models over time, and the neural-adjoint method.Advances in Neural Information Processing Systems, 33:38–48, 2020

work page 2020

[28] [28]

B. Si, Z. Li, Y. Yang, L. Qiao, E. Liu, G. Xiao, and X. Shu. Dynamic indentation testing and characterization of metals based on the split Hopkinson pressure bar (SHPB) device. Mechanics of Materials, 177:104550, 2023

work page 2023

[29] [29]

Simo and T

J. Simo and T. Laursen. An augmented lagrangian treatment of contact problems involving friction.Computers & Structures, 42(1):97–116, 1992

work page 1992

[30] [30]

Svanberg

K. Svanberg. The method of moving asymptotes - a new method for structural optimiza- tion.International Journal for Numerical Methods in Engineering., 24(October 1985):359–373, 1987

work page 1985

[31] [31]

Wriggers.Computational Contact Mechanics

P. Wriggers.Computational Contact Mechanics. Springer Berlin Heidelberg, 2006

work page 2006

[32] [32]

Wriggers, J

P. Wriggers, J. Schr¨ oder, and A. Schwarz. A finite element method for contact using a third medium.Computational Mechanics, 52, 10 2013

work page 2013

[33] [33]

C. Xiao, Y. Deng, and G. Wang. Deep-learning-based adjoint state method: Method- ology and preliminary application to inverse modeling.Water Resources Research, 57(2):e2020WR027400, 2021. A Numerical method for the governing equations We detail the finite element discretization and numerical scheme we use to solve the forward prob- lem. We consider a quad...

work page 2021