A data-driven approach for modeling tension–compression asymmetric material behavior: numerical simulation and experiment

Abstract

In this paper, a direct data-driven approach for the modeling of isotropic, tension–compression asymmetric, elasto-plastic materials is proposed. Our approach bypasses the conventional construction of explicit mathematical function-based elasto-plastic models, and the need for parameter-fitting. In it, stress update is driven directly by a set of stress–strain data that is generated from uniaxial tension and compression experiments (physical). Particularly, for compression experiments, digital image correlation and homogenization are combined to further improve modeling accuracy. Two representative tension–compression asymmetric materials, titanium alloy TC4ELI and high-density polyethylene, are chosen to illustrate the effectiveness and accuracy of our proposed approach. Results indicate that our data-driven approach can predict the mechanical response of elasto-plastic materials that exhibit tension–compression asymmetry, within the small deformation regime. This data-driven approach provides a practical way to model such materials directly from physical experimental data. Our current implementation is limited, however, by a small reduction to computational efficiency, when compared to typical function-based approaches. Moreover, our present formulation is focused on tension–compression asymmetric elasto-plastic materials that are isotropic.

Appendices

Appendix A: Discussion of the effect of homogenization in data-generation, under compression

In this appendix, the effect of the proposed data-driven approach, with and without, adopting our homogenization method for data generation is shown in Sect. 2.1. In our data-driven approach for TC asymmetric materials, three strain components of tension and of compression should be obtained. The traditional method of measuring the compressive strain of a cylinder is to attach strain gauges on the surface of the cylinder. There are two shortcomings for this method. The first is that the deformation range that can be measured is limited. The other is that the deformation of the cylinder surface is itself not uniform. As a result, the measured strain components are not accurate. The 1D data generated using our proposed homogenization method is instead shown in Fig. 7, while the 1D data generated using strain gauges, without applying homogenization, is shown in Fig. 17. The 1D data sets in both Figs. 7 and 17 are used in our proposed data-driven approach to simulate a three-point bending experiment. The geometric model and boundary conditions for the simulation are given in Fig. 10b. Here, only the result of $l = 40$ mm is presented. The reaction force vs. the imposed displacement is plotted in Fig. 18. It can be seen that the proposed approach, using the data generated under compression by homogenization, is closer to experiment. As such, it is preferable to adopt our homogenization approach, as we have done in the corpus of our manuscript.

Table 1 Parameters for generation of 1D numerical uniaxial tension and compression non-dimensional data set using the Drucker–Prager model for isotropic hardening

Appendix B: Discussion on the computational cost of the data-driven approach

To compare the computational efficiency of our proposed data-driven approach and a benchmark continuum model, a 3D model of beam bending is herein considered. The geometry of the beam, and its boundary conditions are shown in Fig. 20a. The left of the beam is held fixed, and a uniform displacement (defined as V) is applied on the right surface, in the y-direction.

This problem is first solved using 1D data numerically generated through the Drucker–Prager model, both under uniaxial tension and compression, as shown in Fig. 19. The material parameters for Drucker–Prager are listed in Table 1.

The reaction force (defined as $F_R$) and the displacement predicted by our data-driven approach and the Drucker Prager model seem in good agreement, as shown in Fig. 20b. Figure 21 further plots the effective stress contours of the beam, under different levels of the applied displacement, $5.0\times 10^{-3}$, $2.6\times 10^{-2}$ and 1.02 mm, marked as I, II and III in Fig. 20b. The computational time (e.g. wall-clock time) of our data-driven approach is 312 s, while that of the Drucker Prager model is only 12 s. These times are for an Intel Core i7-7700 CPU. Clearly, our data-driven approach runs slower than the benchmark model, largely as a result of its calculation of derivatives of stress with respect to strain, for the tangent stiffness formulation based on pure data. Nonetheless, we reiterate here that the anticipated time saving from our data-driven approach does not lie in the simulation time itself, but in its complete by-pass of material model selection, and material parameter calibration, opening a new door for fully automated material modeling in product development cycles.