Learning viscoplastic constitutive behavior from experiments: II. Dynamic indentation

TL;DR

Extends the PDE-constrained inverse problem framework from Part I to dynamic indentation with unilateral contact, formulating a forward model that combines small-strain J2 viscoplasticity with rate dependence and a contact law enforced via a Lagrange multiplier and slack variable. Adjoint-based sensitivities are derived to efficiently identify constitutive parameters by matching measured reaction forces and indenter displacements to model predictions, solved with a staggered time-stepping scheme. Demonstrations on synthetic data and dynamic indentation experiments for RHA steel and Al 6061-T6 show accurate recovery of parameters such as $\sigma_y$, $\varepsilon_0^p$, $n$, $\dot{\varepsilon}_0^p$, and $m$, with validation against independent uniaxial tests and discussion of model assumptions influencing elastic estimates. The work highlights the richness of dynamic force fluctuations for parameter recovery and outlines a path toward generalized constitutive laws using neural operators in Part III.

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.

Figures (7)

• Figure 1: Schematic diagram of the rigid indentation test. We consider a cylindrical domain $\Omega$ of radius $R_C$ and height $h$ resting on a rigid surface indented on its top surface $\partial_C \Omega$ by a rigid sphere of radius $R_I$ to a prescribed depth $\delta$.
• Figure 2: Demonstration with synthetic data for the the data generated with $P^{\text{gen}}_1$, with the optimization initialized from $P^{\text{init}}_1$. (a) (i) Filtered force vs indentation depth for the synthetic, initialized, and learned parameters for the four indenter velocities tested. (a)(ii) The value of the hardening variable at the final timestep for the indentation tests at the contact site for the synthetic and learned parameters sets. For visualization purposes, these were conducted on a mesh with one level more refinement than the optimization was conducted on. (b) Normalized objective vs iteration. (c) Comparison of synthetic and learned models in an independent test of uniaxial tension at various strain rates.
• Figure 3: Response to an independent uniaxial tensile test using the synthetic parameters $P^{\text{gen}}$ and the converged parameters. (a) Response for $P^{\text{rec}}_2$. (b) Response for $P^{\text{rec}}_3$.
• Figure 4: Demonstration on linearized synthetic data generated from $P^{\text{gen}}_\text{lin}$. (a) Filtered forces vs indentation depths for the original synthetic parameters (solid grey lines), linearization (solid black lines), initialized (blue dashed lines), and learned (dashed red lines) parameters for the four constant indenter velocities tested.(b) Normalized objective vs iteration. (c) Comparison of synthetic and learned models in an independent test of uniaxial tension at various strain rates.
• Figure 5: Results for experiments with RHA steel. (a ) Indentation velocity vs time for the two tests conducted. (b) Force versus indentation depth for (i) velocity profile 1 and (ii) velocity profile 2. (c) Normalized objective versus iteration. (d) Comparison of the recovered model with two literature models in independent uniaxial tension tests conducted at room temperature at a strain rate of $5 \times 10^3$/s.