achieved overall specimen profile as confirmation of constitutive model validity. ... Mathematica [15] and Python [16] scripts have been written to automate the ...
INL/CON-10-19255 PREPRINT
Optimization-Based Constitutive Parameter Identification from Sparse Taylor Cylinder Data 81st Shock and Vibration Symposium J. M. Lacy J. K. Shelley J. H. Weathersby G. S. Daehn J. Johnson G. Taber October 2010 This is a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint should not be cited or reproduced without permission of the author. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, or any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for any third party’s use, or the results of such use, of any information, apparatus, product or process disclosed in this report, or represents that its use by such third party would not infringe privately owned rights. The views expressed in this paper are not necessarily those of the United States Government or the sponsoring agency.
OPTIMIZATION-BASED CONSTITUTIVE PARAMETER IDENTIFICATION FROM SPARSE TAYLOR CYLINDER DATA J. M. Lacy1, J. K. Shelley1, J. H Weathersby1, G. S. Daehn2, J. Johnson2, and G. Taber2 ABSTRACT The classic Taylor impact test imparts temporally and spatially varying fields of strain, strain rate, and temperature through the specimen. It is possible to exploit this complexity to directly identify constitutive model parameters from the deformed shape of the specimen. Where prior investigators have employed various mathematical fitting methods to identify or improve strength model parameters from Taylor cylinder profiles, we extend the method to employ a multi-objective genetic optimization algorithm to minimize the cylinder profile errors simultaneously on three cylinders impacted at different velocities. No experimental data other than the three Taylor cylinders is employed in developing the constitutive model parameter set, and generic starting coefficients are employed. To validate the accuracy of the resulting coefficients, both split Hopkinson pressure bar and axisymmetric expanding ring tests were conducted and compared to the resultant Johnson-Cook strength model. The derived strength model agreed well with experimental data available to date. Further work is necessary to evaluate the range of rates and temperatures over which parameters derived by this method may be applied.
INTRODUCTION The Taylor impact test is historically a popular method for verifying constitutive models and identifying a subset of model parameters, supplementing data provided by other experimental methods. Due to the complexity of the strain, strain-rate, and temperature fields present in a Taylor specimen, it is possible to identify material strength parameters dependent on those fields directly from Taylor data. Such an approach could reduce the quantity of testing required to characterize materials for input into computational analysis of ballistic or blast events, enhancing analysis timeliness and accuracy. Parameter identification methods typically compare measured deformation of the cylinder at a restricted number of locations to the deformations predicted by a hydrocode. For example, Johnson and Holmquist [1] measured final length, maximum diameter, and one intermediate diameter to obtain the three yield and strain hardening constants (A, B, and n) of the Johnson-Cook Model. Allen, Rule, and Jones [2] extended the method by employing a secondorder polynomial response surface to minimize the difference between the experimental and computed cylinder profile at a single impact velocity. They had greatest success when beginning with parameter values calibrated to quasi-static test data. In the current work, we extend the method to employ a multi-objective genetic optimization algorithm to minimize the cylinder profile errors simultaneously on three cylinders impacted at different velocities. No test data other than the three Taylor cylinders is employed and we used a non-representative set of starting coefficients to challenge the optimization algorithm. To ensure that the resulting coefficient set is applicable to other load states, both split Hopkinson pressure bar and axisymmetric expanding ring tests were conducted and compared to the resultant Johnson-Cook model. Satisfactory agreement was obtained.
TAYLOR IMPACT TEST The test was first developed by Taylor [3] as a method to determine the dynamic yield stress of idealized materials. The test specimen is a right circular cylinder of the subject material launched into a theoretically rigid target. After the shot, the specimen is recovered and its final shape, including overall length and diametric profile, is carefully measured, along with any fractures that may have occurred. The final deformed shape and fracture pattern may then be used to develop or refine the constitutive and failure models for the material.
1 2
Idaho National Laboratory, Idaho Falls, ID Ohio State University, Columbus, OH.
Taylor derived a method to determine the dynamic yield stress from measurements of initial velocity, impact velocity, and undeformed specimen length. For the sake of brevity, the derivation is not presented here, however an excellent presentation is made by Zukas [4]. Taylor’s 1-dimensional derivation has been modified and extended several times to include elastic strains [6], non-linear work hardening [7], non-rigid targets, and failure loci [8]. Several authors, for example Holt et al. [9] and Lapczyk et al. [10] compare experimental and computationally achieved overall specimen profile as confirmation of constitutive model validity. It is challenging to identify constitutive model parameters directly from Taylor data because the stress, deformation, and temperature fields in the specimen vary both temporally and spatially throughout the impact event. Plastic strain varies from well over 1.0 at the impact surface to near zero at the rear surface. Similarly strain rate and temperature vary strongly with both axial and radial position in the specimen. This complexity, which makes direct identification of strength parameters impossible, is also a great attraction. Fewer experiments should be necessary to evoke the range of desired stress/strain/temperature states. Similarly, parameter sets derived from Taylor data could be expected to apply over the entire range of strain, rate, and temperatures evoked by the basis tests.
JOHNSON-COOK STRENGTH MODEL There are a broad range of advanced constitutive models available for use in modern hydrocodes. For several reasons, we chose to work with the classic Johnson-Cook [11] model in this effort. It is reasonably simple, employing just 5 parameters to identify. It is ubiquitous, supported by all computational dynamics codes of which we are aware, and it is phenomenological, not requiring a microstructural basis for parameter values. Thus it lends itself well to an automated parameter identification process that has no knowledge of the internal structure of the material being evaluated. The Johnson-Cook constitutive model is expressed as
Vy
·
§
A BH ¨¨1 C ln HH ¸¸ 1 T n p
©
ref
*m
¹
(1)
Where y is the yield stress, p is the effective plastic strain, H p is the plastic strain rate, and H pref is the reference plastic strain rate. A, B, and n are yield and strain-hardening parameters, C is a rate-hardening parameter, and m is a thermal softening parameter. We use a reference plastic strain rate of 1.0s-1, with Schwer’s [12] cautions regarding the reference strain rate duly noted. T*, the homologous temperature, is variously defined as
T*
T Troom Tmelt Troom
(2)
as originally proposed by Johnson and Cook or as
T*
(3)
T Tmelt
as proposed by Gray, et al [13]. Here we use the original form.
EXPERIMENTAL APPROACH We chose A36 structural mild steel from 2” thick rolled plate as the subject material for this investigation. Because it is widely available and inexpensive it is often used as a baseline or witness material in blast and ballistic studies. However, because it is never used directly in blast- or penetration-resistant systems, it is not well characterized in this load regime. It is anticipated that analysts performing baseline or witness penetration computational studies might benefit from characterization of this material.
Cylinders of dimension 1.27 cm diameter by 4.23 cm length were impacted into a massive anvil of AR400 steel, at 3 velocities of 249.4, 269.4, and 284.5 m/sec. Figure 1 shows a typical specimen rebounding from the anvil face at 250 μsec after impact. Rebound is observed to begin at approximately 100 μsec after touchdown. Figure 2 shows the AR400 anvil face after several impacts. Note that the anvil face is observably deformed by each specimen, demonstrating that the usual assumption of a perfectly rigid surface is not entirely valid. We chose to incorporate an approximation of the elastic-plastic response of the anvil to enhance accuracy of the simulations, though we did not evaluate the improvement in fidelity with respect to the rigid face assumption.
Figure 1. Cylinder rebounding from anvil face
Figure 2. Plastic deformation of AR400 anvil face. Ball-point pen for scale.
To measure the profile of deformed cylinders, a commercially available desktop 3-dimensional laser profilometer system was employed. The NextEngine [14] laser scanner provides a full-volume description of the deformed specimen, within a tolerance of 0.13mm (0.005”), which can be imported into a variety of solid modeling packages. The full volume description allows us to quickly compare profiles around the specimen to evaluate ovality (material anisotropy) or asymmetry of the impact, and select the most appropriate profiles for comparison to computational results. Mathematica [15] and Python [16] scripts have been written to automate the profile generation process.
MULTI-OBJECTIVE GENETIC OPTIMIZATION METHOD We employed the DAKOTA toolkit [17] as the framework for the optimization study. Our target was a parameter set that minimized the error between computed and observed cylinder profiles simultaneously for three specimens impacted at different velocities (thus multi-objective). A genetic algorithm approach was selected to modify and evaluate the fitness of candidate parameter sets with 5 design variables (A, B, n, C, and m). 50 generations were allowed, with a maximum of 2500 simulations. A mutation rate of 0.1 was employed. An impact simulation was performed for each parameter set with the Eulerian finite-difference code CTH V9.1[18]. 2-dimensional axisymmetry was employed, but the usual assumption of a perfectly rigid impact face was discarded as unnecessarily simplistic. Rather, the anvil response was approximated using a Johnson-Cook plasticity model representative of rolled homogeneous armor steel. The domain was meshed with uniform cells of dimension 0.1mm x 0.1 mm, and all runs carried out to 160 μsec to ensure completion of the impact event without operator intervention. To start, we created a non-representative parameter set with yield and hardening coefficients representative of a fairly hard material, with the expectation that the optimization method should be able to overcome large errors in the initial guess.
We automated the optimization process with a few scripts to set up the run parameters and directory structures, and to post process the results. For each parameter set, three simulations were performed to capture the three impact velocities. Each simulation took around 10 minutes to run using 128 cores. Table 1. Starting Johnson-Cook Parameters A B (MPa) (MPa) 996 391
n
C
m
0.31
0.025
1.09
Each computed profile was compared to its corresponding experimental profile for the given velocity. Comparison points were evenly spaced at 0.38 mm (0.015 in.) on the vertical axis for a total of 62 to 75 points, depending on final cylinder height. We calculated the RMS sum error for all comparison points and returned that value to DAKOTA as input to the next set of designs. The optimization was allowed to proceed until the returned profile error value did not change more than 5% for 5 generations. Table 2 presents the resulting parameter set for the Johnson-Cook strength model for A36 steel alloy. These values deviate significantly from the starting numbers in the initial yield (A), strain hardening (B), and thermal softening (m), with relatively less shift in the strain hardening exponent (n) and strain rate coefficient (C).
Table 2. Optimized Johnson-Cook Parameters for A36 Mild Steel A B n (MPa) (MPa) 146.7 896.9 0.320
C
m
0.033
0.323
VALIDATION Given that this parameter set was derived from only 3 experiments, we must investigate the extent to which the results are applicable to other load states independent of those obtained via the Taylor cylinder. To that end, specimens from the same plate of steel were subjected to two other load conditions: the split Hopkinson pressure bar (SHPB) and the axisymmetric expanding ring (AER) tests. Data from both are compared to the final Taylorderived parameter set as a way to check the validity of the parameter set. We did not incorporate these data into the parameter extraction process in order to maintain independence of the reference data from the measured quantities. SPLIT HOPKINSON PRESSURE BAR The INL split Hopkinson pressure bar (SHPB) is set up in the classic fashion. Excellent presentations of the apparatus, method, and derivation of stress-strain relationships from the output signals are presented by Zukas et al [19], Gray [20], and Kaiser [21]. The INL apparatus employs Maraging 300 steel incident and transmitted bars 36.0 in. long by 0.498 in. diameter with a matching striker bar 15.3 cm in length. Specimens for this test series were nominally 4.75mm in thickness and 9.5 mm diameter. Data was recorded by a National Instruments high-speed digitizer system (PXI-based). Signal analysis is performed using locally developed Python and Labview [22] scripts. Two specimens were shot at room temperature and strain rates of 3500 s-1 and 7500 s-1, and a third specimen shot at a temperature of 425K and strain rate of 1700 s-1. AXISYMMETRIC EXPANDING RING The axisymmetric expanding ring test is being jointly investigated by the INL and Ohio State University as an alternative high-strain-rate material characterization method. Based on the rapid compression or expansion of an instrumented axisymmetric ring, the technique employs electromagnetic forces or a small exploding bridge wire as the motive force; photon Doppler velocimetry to measure radial displacement, velocity and acceleration; and automated software systems to objectively develop stress-strain relationships from measured parameters. The uniform, high-strain-rate and large tensile strains imposed on the specimen are distinctly different from the stress and strain fields imposed by compressive impact tests such as the Taylor and SHPB methods. When true axisymmetry is obtained, stress waves travel only radially through the specimen thickness, while the tangential
stress being measured is spatially uniform. In contrast, loading and relaxation waves reflect back and forth through both Taylor and SHPB specimens during the deformation event, adding complexity to the analysis. Four rings measuring 50.8 mm ID x 3mm tall x 1 mm thick were expanded in the OSU apparatus pictured in Figure 3. An exploding bridge wire in a urethane cylinder was used to drive the expansion, thus there was no ohmic heating of the specimens, which remained near room temperature. Results of these shots are reported in Table 2. Reduction of the displacement-velocity history data to extract the stress-strain path of the specimen is as yet incomplete. At this time we must report only the average yield stress of the 4 specimens of 425 MPa.
Figure 3, AER specimen mounted on urethane driver block, with exploding bridge wire driver.
Table 2. Axisymmetric Expanding Ring Results for A36 Steel ID Exp. Energy (kJ) Pk. strain rate Max. Strain (%) (%) 3200J RD 4000J RD 4400J RD 4800J RD
3.2 4.0 4.0 4.8
1830 2138 2441 2898
25.5 27.0 32.1 32.4
Fracture
No No 3 pcs 5 pcs
Average Yield stress (MPa)
425
RESULTS True stress-true strain relationships derived from the extracted J-C parameters are plotted in Figure 4 along with experimentally-obtained data. Quasi-static tensile data for generic A36 material is taken from Fisher and Iwankiw [23]. Compressive SHPB and tensile AER data are from the current effort as described herein. In the Figure, the JC model stress strain relationship is plotted assuming isothermal response for strain rates below 1000s-1, and adiabatic response for rates above that. Adiabatic temperature increase is taken as
T
D
³ Uc
V dH pl
(4)
p
Where and cp are the sample density and specific heat, is the fraction of plastic work converted to thermal energy (usually taken as 0.9), and and pl are the deviatoric stress and effective plastic strain, respectively.
We observe satisfactory agreement between the derived J-C parameter set and the SHPB experimental data at both room temperature and 425K. The average yield stress observed from the AER tests, while in agreement with the other data, is insufficient to aid in confirming the validity of the model. While we do not yet have enough data to confirm the accuracy of the derived thermal softening parameter, the result at 425 K is encouraging. Because the volume of the high-temperature region in the Taylor cylinder specimen is relatively small, we consider accurately capturing the thermal response the greatest challenge of this approach.
CONCLUSIONS The classic Taylor impact test imparts temporally and spatially varying fields of strain, strain rate, and temperature through the specimen. It is possible to exploit this complexity by identifying constitutive model parameters that predict the final deformed shape of the specimen. We employed a multi-objective genetic optimization method to identify the best-match parameter set for a group of three specimens impacted at different velocities. No other experimental data was used to calibrate or guide the optimization process. We find the method provides acceptable agreement with experimental split Hopkinson pressure bar compression and axisymmetric expanding ring tensile results available to date. In an ongoing study, we are applying the method to other materials with different strain response and ductilities —Ti-6Al-4V, OFC copper, and high-strength, low-alloy steel—to assess the breadth of the method’s applicability. Further work is necessary to evaluate the range of rates and temperatures over which parameters derived by this method may be applied.
1,000
900
800
700
TrueSterss(MPa)
600
500
400 JCModel,RT,0.001s1 JCModel,RT,3500s1 300 JCModel,RT,7500s1 JCModel,425K,1700s1 200
Tensile,quasistatic SHPB,RT,3500s1 SHPB,RT,7500s1
100
SHPB,425K,1700s1 AxisymmetricRing,averageyield 0 0.00
0.05
0.10
0.15
0.20
TrueStrain
Figure 4. Comparison of Taylor-extracted J-C model coefficients to experimental SHPB and AER data.
0.25
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