Simulation and dimensional analysis of instrumented dynamic spherical indentation of ductile metals
Graphical abstract
Introduction
Dynamic instrumented indentation is the focus of the present investigation. Unlike static indentation, dynamic indentation offers the possibility of characterization of the strain rate sensitivity of the material [1], [2], [3], [4], [5]. Furthermore, inertia and temperature changes under near-adiabatic conditions could affect results, the former especially at very high impact velocities [6], [7]. In dynamic indentation, strain rate, like strain itself, is also highly nonuniform, even if the indenter’s velocity is constant [6]. In any real experiment, the indenter will always decelerate prior to unloading.
Instrumented dynamic indentation using the split Hopkinson pressure bar (SHPB) (i.e., Kolsky bar), as often first credited to Subhash and co-workers [8], [9], has been invoked with a variety of indenter geometries and target materials. Instrumented dynamic indentation enables recording of velocities and mechanical forces during the transient indentation process for subsequent analysis. Recent methods with the SHPB have used a full sphere sandwiched between two specimens [10] or a striker with variable impedance to achieve load cycling [11], [12], [13]. Dynamic conical indentation experiments were used [2] to deduce strain rate sensitivity, where the indenter was propelled by a light gas gun, and a combination of interferometry and load transducer was used to obtain a time-resolved response. A novel aspect of the current study, compared to other works that simulate instrumented dynamic indentation [2], [10], [14], is the present use of a miniaturized SHPB rather than a full-scale system. Advantages of miniaturization include reduced inertial effects and decreased velocity rise times in the input and output bars [13], [15]. Samples are generally smaller in size, and higher strain rates can be achieved through miniaturization, at least for conventional (uniform versus indentation) loading [15].
Since the end goal of the present study is establishment of a combined experimental–numerical methodology to extract dynamic properties, some prior prominent works on this topic are noted. Burley et al. [16] performed dynamic 5 mm-diameter spherical impact experiments and FE simulations on copper targets, with impact velocities ranging from 50 to 300 m/s. Strain rate sensitivity and frictional properties were obtained by a goodness-of-fit procedure on displacement–time histories and residual indent shapes. Characteristic strain rates ranged from 10 to 10/s, and temperature rises were significant enough to impart thermal softening. Ito et al. [4] performed dynamic FE simulations of ballistic impact of spheres into 10 different target metals to verify an analytical expression relating crater depth to strain rate sensitivity of flow stress. Impact velocities ranged from 0.6 to 180 m/s, and strain rates from 10 to 10/s. That method [4] for identification of rate sensitivity requires indentation depths attained from impacts at two distinct velocities. Lu et al. [2] used dynamic sharp indentation gas-gun experiments, verified by FE modeling, to obtain strain rate sensitivity of copper. Impact velocities ranged from 6 to 35 m/s and strain rates from 875 to 1750 m/s. Dynamic plasticity model parameters were obtained from simulations of specimens in full-size Kolsky bar tests on steel in tension at rates from 500 to 1500/s [17]; simulations of these and other full-scale Kolsky bar experiments at up to 3600/s were used to obtain dynamic properties for two steels in [18]. Nguyen et al. [19], [20] obtained yield, hardening, and rate sensitivity parameters for steels using indentation and FE modeling at lower strain rates ranging from 0.002 (quasi-static) to 0.4/s (low-rate dynamic).
In the current work, the dynamic dimensional analysis [7], [21], [22] is further refined to efficiently study effects of parameters entering the popular Johnson–Cook plasticity model [23]. This is one of the most widely used high-rate constitutive model for ductile metals because of its simplicity and containment of necessary physical ingredients to curve-fit experimental stress–strain data at different strain rates and temperatures. The current research directly models instrumented dynamic spherical indentation experiments in a miniature Kolsky bar apparatus, where the experiments were recently designed and implemented by Casem and co-workers [11], [12], [13]. The first stage of the investigation assesses suitability of the explicit dynamic FE code ALE3D [24] to reproduce the experimental test configuration and the data acquired from three different complex applied velocity histories, rather than constant indenter velocities which are much more easily implemented numerically [6], [7] but impossible to achieve in real experiments. To this end, accuracy of the Johnson–Cook model enters the procedure, where representative baseline properties from the literature [15], [25], [26], [27] on the substrate material are used, without attempting calibration to the current indentation data. The material of choice is a polycrystalline aluminum (Al) alloy designated Al 6061-T6. Mesh densities necessary for extraction of transient indentation force-depth and stress–strain data, the latter requiring the contact radius, are determined. Local field variables (e.g., stress, plastic strain, temperature) and global variables such as mean contact pressure are elucidated.
The second major stage of the current investigation establishes the framework for dimensional analysis and applies this framework to systematically investigate relative effects of constitutive parameters (as well as initial temperature) on the predicted dynamic indentation response. Results show which parameters may be reasonably expected to influence indentation force-depth data, and thus are candidates for extraction via calibration of FE solutions to experiments. Implementation of such calibration methods, which likely requires sophisticated numerical optimization strategies (e.g., machine learning techniques [28], [29] for accuracy, speed, and stability) as well as analysis of uniqueness of inverse solutions [19], [30], [31], is outside the scope of the present study.
The third and final stage of this investigation demonstrates successful extraction of the strain-rate sensitivity parameter from experimental data and parametric FE simulations. An error measure associated with cumulative work of indentation is defined and minimized for parameter optimization. Stress–strain predictions of the constitutive model and parameter set are compared versus those obtained from external studies that used conventional static and dynamic methods (e.g., standard SHPB) and are also validated versus static indentation data from an independent experimental source.
Sections of this paper include the following content. Experiments [13], [22] are described in Section 2, followed by more extensive details of the FE model (which has never been presented elsewhere), with baseline parameters. Results of Section 2 reveal both success and limitations of the baseline model for reproducing certain features observed in data from three instrumented dynamic tests. The dimensional analysis and sensitivity investigation are discussed in Section 3, including proposed experimental methods for extraction of parameters for previously uncharacterized ductile metals. Extraction of strain rate sensitivity and parameter comparisons are presented in Section 4. Conclusions follow in Section 5. Exploratory simulations including much higher loading rates are contained in Appendix A. Background on indentation fundamentals and supplementary figures are included in the on-line electronic supplementary material (SM).
Section snippets
Experimental protocols
Experimental methods have been discussed by Casem [11], [12], [13] and are summarized in what follows. A miniature Kolsky bar (i.e., SHPB) [15] is adapted for instrumented dynamic spherical indentation. Transient force, displacement, and velocity data are acquired or inferred from each experiment. The loading history (e.g., indenter’s velocity) depends on the velocity of the striker bar and geometric properties of the system (including pulse shaping) as well as indentation resistance due to
Dimensional analysis and parameter sensitivity
Given a sample of solid material with unknown physical properties, a typical objective of indentation experiments is determination of such properties via analysis of load-depth history data, as well as analysis of possibly available information on contact radius and the size and shape of any residual imprint. If functional relationships between the indentation response and constitutive properties are available for a given set of boundary and initial conditions, then it may be possible to invert
Johnson-Cook parameter determination
An ultimate goal of the combined experimental–numerical approach is identification of elastic, plastic, and other constitutive properties of the indented material. Results in Section 3 show that at modest input velocities (order of one to several m/s) and initial room temperature (not elevated temperature) conditions, the load depth ( vs. ) response is sensitive to elastic constants and the Johnson–Cook yield and hardening parameters . A brute-force calibration of all 6 elastic and
Conclusions
Simulations of novel instrumented dynamic indentation experiments in a miniature Kolsky bar have been performed. The tested material is aluminum alloy Al 6061-T6. The FE model, with Johnson–Cook plasticity used for the alloy, reasonably replicates the curvature of the experimental load versus depth data for three different experimental loading histories corresponding to different velocity profiles for the Kolsky input bar. Effective strain rates for simulations of these three experiments,
CRediT authorship contribution statement
J.D. Clayton: Concept, Design, Analysis, Writing, Revision of the manuscript. J.T. Lloyd: Concept, Design, Analysis, Writing, Revision of the manuscript. D.T. Casem: Concept, Design, Analysis, Writing, Revision of the manuscript.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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