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Research Papers

On the Extraction of Elastic–Plastic Constitutive Properties From Three-Dimensional Deformation Measurements

[+] Author and Article Information
A. J. Gross

Center for Mechanics of Solids,
Structures, and Materials,
Department of Aerospace Engineering and
Engineering Mechanics,
University of Texas at Austin,
Austin, TX 78712-1221

K. Ravi-Chandar

Fellow ASME
Center for Mechanics of Solids,
Structures, and Materials,
Department of Aerospace Engineering and
Engineering Mechanics,
University of Texas at Austin,
Austin, TX 78712-1221
e-mail: ravi@utexas.edu

This restriction arises from the desire to use measurements in the wavelengths at which the specimen is opaque; in principle, the use of X-ray tomography [[12]], laminography [[13]], and other tools could provide information on the interior, but such methods are still under development, limited to identifying damage, experimentally expensive and await further development.

The authors would like to thank Dr. B.L. Boyce of Sandia National Labs for performing the heat treatment.

Similar specimens have been used by other researchers: see Tardiff and Kyriakides [14] and Boyce et al. [16] for recent examples of such use.

It is common to use the square root of the sum of squared error (L2 norm), with the underlying assumption that the errors are from random fluctuations in the experimental quantities and therefore could be idealized as Gaussian distributed. Here, systematic errors in the numerical solution dominate the random experimental fluctuations so we have taken the absolute value (L1 norm) for the error. While the L2 norm is dominated by large errors, the L1 norm accumulates all errors uniformly.

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 19, 2014; final manuscript received March 24, 2015; published online June 3, 2015. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 82(7), 071013 (Jul 01, 2015) (15 pages) Paper No: JAM-14-1590; doi: 10.1115/1.4030322 History: Received December 19, 2014; Revised March 24, 2015; Online June 03, 2015

In this article, a coupled experimental and numerical method is utilized for characterizing the elastic–plastic constitutive properties of ductile materials. Three-dimensional digital image correlation (DIC) is used to measure the full field deformation on two mutually orthogonal surfaces of a uniaxial tensile test specimen. The material’s constitutive model, whose parameters are unknown a priori, is determined through an optimization process that compares these experimental measurements with finite element simulations in which the constitutive model is implemented. The optimization procedure utilizes the robust dataset of locally observed deformation measurements from DIC in addition to the standard measurements of boundary load and displacement data. When the difference between the experiment and simulations is reduced sufficiently, a set of parameters is found for the material model that is suitable to large strain levels. This method of material characterization is applied to a tensile specimen fabricated from a sheet of 15-5 PH stainless steel. This method proves to be a powerful tool for calibration of material models. The final parameters produce a simulation that tracks the local experimental displacement field to within a couple percent of error. Simultaneously, the percent error in the simulation for the load carried by the specimen throughout the test is less than 1%. Additionally, half of the parameters for Hill’s yield criterion, describing anisotropy of the normal stresses, are found from a single tensile test. This method will find even greater utility in calibrating more complex material models by greatly reducing the experimental effort required to identify the appropriate model parameters.

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References

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Figures

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Fig. 1

A generic BVP indicating the region of observation for acquiring additional kinematic measurements for use in the IP

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Fig. 2

(a) Drawing of the tensile specimen with dimensions in mm. Note the small curvature over the gauge section and square cross section. (b) Orientation of the tensile specimen relative to the natural directions of texture in the sheet. This is also the perspective that the specimen is viewed from by the stereo imaging system to capture deformation information on two orthogonal surfaces. The shaded area is the region observed by the imaging system.

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Fig. 3

Nominal stress–strain curve for 15-5 PH in the H-1075 condition. Strain was measured with a DIC-based extensometer, completely spanning the necked region. The first load peak occurs at a strain of about 0.9%, the minimum is reached at a strain of about 1.4%, and the onset of necking occurs at a strain of about 7.6%. Full field strain contours are shown in Fig. 4 at the points marked (a)–(d).

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Fig. 4

Spatial variation of the maximum true (logarithmic) principal strain field ɛ1(x) as measured from 3D-DIC at stages (a)–(d) marked in Fig. 3. The front surface is the one below the corner and the side surface above. The white spaces are where DIC failed to correlate due to proximity to the specimen’s corner or local defects in the speckle pattern.

Grahic Jump Location
Fig. 5

(a) Spatial variation of the true (logarithmic) strain field ɛ1(x) as measured from 3D-DIC just prior to rupture. The maximum true strain measured is around one. The small black and red (color online) dots indicate locations along the midline of each surface, where the spatial variation of strain is plotted in Fig. 5(b). The two large black dots are located at the center of necking. At these locations the stress state remains approximately uniaxial throughout the test. Strain data from these two points are used in Figs. 6 and 7. (b) Spatial variation of the longitudinal and transverse strains on both surfaces plotted at 30 s intervals for the last 3 min of the test (nominal strain values of 0.106, 0.117, 0.129, 0.142, 0.155, and 0.169). The longitudinal strains on both surfaces remain nearly identical to one another through the duration of the test. Transverse strains accumulate more rapidly on the side surface than the front, indicating a lower stiffness in the short transverse direction than long transverse.

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Fig. 6

Variation of the transverse strain with respect to longitudinal strain at the deepest point in the neck on the front and side surfaces. The transverse strains develop nonlinearly with continued longitudinal straining, thus indicating that the anisotropy may be evolving throughout the test. Evolution appears to be the most rapid around a longitudinal strain of 0.14, where the curvature of the lines reaches their maximum magnitude.

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Fig. 7

The transverse strains on both surfaces plotted against each other. The variation is nearly linear, with the slope of the line providing one of Lankford’s parameters as 0.895. When investigated closely, the Lankford parameter is not constant; it varies most rapidly at the low strain range and then settles to a nearly constant value with increasing strain. This behavior is discernable from the measurements taken in this experiment, but the level of uncertainty is high to investigate this behavior closely.

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Fig. 8

Spatial discretization used for the FE model. Two of the three symmetries used are visible. Necking occurs at the symmetry plane on the right, where the refined mesh is located. The smallest mesh dimension is 76 μm.

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Fig. 9

Objective function value for the best member of each generation throughout the global optimization procedure. The curve is flat when a lower value for the objective function has not been found from the previous generation.

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Fig. 10

Comparison of the nominal stress–strain curves produced by both objective functions to the experimental observation. Both of the simulated curves closely follow the experimental result.

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Fig. 11

Spatial distribution of the displacement error for the load optimized material just before rupture. The front surface is the lower area and the side surface is the upper area. The center of the neck is located at x = 0. The region shown is not the entire surface, but corresponds to x < 0, y > 0, and z > 0. Error is relatively uniform across both surfaces within and beside the necked region.

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Fig. 12

Spatial distribution of the displacement error for the displacement optimized material just before rupture. The front surface is the lower area and the side surface is the upper area. The center of the neck is located at x = 0. The region shown is not the entire surface, but corresponds to x < 0, y > 0, and z > 0. Error tends to be lower on the front surface and varies along the axis of the specimen.

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Fig. 13

Spatial (x-axis) and temporal (y-axis) variation of the displacement error along a line near the midline of the front surface for the load optimized material. Errors that are relatively spatially uniform increase most rapidly during necking.

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Fig. 14

Spatial (x-axis) and temporal (y-axis) variation of the displacement error along a line near the midline of the front surface for the displacement optimized material. Errors that are relatively spatially uniform increase most rapidly during necking.

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Fig. 15

Out of plane displacement on the midline of the front surface in the necked region. Both load and displacement optimized materials show satisfactory agreement.

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Fig. 16

Out of plane displacement on the midline of the side surface in the necked region. The displacement optimized material matches the experimental result much closer than the load optimized material does.

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Fig. 17

Strain hardening behavior for load and displacement optimized materials. The displacement optimized material is more compliant at high strains. The tangent modulus of the load optimized material is almost 25% stiffer at a (logarithmic) plastic strain of 0.5.

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Fig. 18

(a) Sensitivity of the displacement blind objective function to changes in the model parameters that define anisotropy. Data were produced using the load optimized strain hardening behavior and the values are normalized by the minimum. This objective function is not sensitive to changes in Lankford’s value. (b) Sensitivity of the displacement aware objective function to changes in the model parameters that define anisotropy. Data were produced using the displacement optimized strain hardening behavior and the values are normalized by the minimum. This objective function is sensitive to both parameters that define anisotropy.

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Fig. 19

Variation of the transverse strain with respect to longitudinal strain at the deepest point in the neck on the front and side surfaces. The result from the simulation with the displacement-optimized material parameters are overlaid on the experimental measurement.

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