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Research Papers

Determination of the Shear Modulus of Orthotropic Thin Sheets With the Anticlastic-Plate-Bending Experiment

[+] Author and Article Information
Nengxiu Deng

Department of Mechanical Engineering,
Center for Advanced Materials and
Manufacturing Innovation (CAMMI),
University of New Hampshire,
33 Academic Way,
Durham, NH 03824

Yannis P. Korkolis

Department of Mechanical Engineering,
Center for Advanced Materials and
Manufacturing Innovation (CAMMI),
University of New Hampshire,
33 Academic Way,
Durham, NH 03824
e-mail: yannis.korkolis@unh.edu

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received February 11, 2018; final manuscript received May 11, 2018; published online June 22, 2018. Assoc. Editor: Huiling Duan.

J. Eng. Mater. Technol 140(4), 041011 (Jun 22, 2018) (7 pages) Paper No: MATS-18-1039; doi: 10.1115/1.4040352 History: Received February 11, 2018; Revised May 11, 2018

The shear modulus of orthotropic thin sheets from three advanced high-strength steels (AHSS) is measured using the anticlastic-plate-bending (APB) experiment. In APB, a thin square plate is loaded by point forces at its four corners, paired in opposite directions. It thus assumes the shape of a hyperbolic paraboloid, at least initially. The principal stress directions coincide with the plate diagonals, and the principal stresses are equal and opposite. Hence, at 45 deg to these, a state of pure shear exists. A finite element (FE) study of APB is reported first, using both elastic and elastoplastic material models. This study confirms the theoretical predictions of the stress field that develops in APB. The numerical model is then treated as a virtual experiment. The input shear modulus is recovered through this procedure, thus validating this approach. A major conclusion from this numerical study is that the shear modulus for these three AHSS should be determined before the shear strain exceeds 2 × 10−4 (or 200 με). Subsequently, APB experiments are performed on the three AHSS (DP 980, DP 1180 and MS 1700). The responses recorded in these experiments confirm that over 3 × 10−4 strain (or 300 με) the response differs from the theoretically expected one, due to excessive deflections, yielding, changing contact conditions with the loading rollers and, in general, the breaking of symmetry. But under that limit, the responses recorded are linear, and can be used to determine the shear modulus.

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Figures

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Fig. 1

Sensitivity of orientational dependence of (a) Young's modulus and (b) Poisson's ratio on the value of the shear modulus, (Adapted from Ref. [1])

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Fig. 2

Schematic of the APB experiment (Adapted from Ref. [25])

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Fig. 3

(a) Finite element model for studying the APB experiment; shown are the orientations discussed in the text and, in an inset, the arrangement of the virtual strain-gages. (b) Deformed configuration of the model, showing contours of total displacement. The model shown in (a) has been mirrored twice in (b), for easier visualization.

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Fig. 4

Force–displacement curve during APB of the three steels in this study. Included are experiments and FEA predictions.

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Fig. 5

Predictions of strain evolution in the two strain gages (using the notation of Fig. 3(a)), for three different approaches and material models. At low strains, all approaches agree with each other.

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Fig. 6

Predictions of the shear stress–strain for the three steels. (a) Overview and (b) view zoomed-in around the origin, showing the shear modulus as the slope of this curve.

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Fig. 7

Anticlastic-plate-bending experiment. (a) Photograph of the specimen, also showing the strain-gage arrangement, and (b) schematic of the experiment, indicating the biaxial stresses that develop.

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Fig. 8

Experimentally determined strain history from the three strain gages installed on a DP 980 specimen. The gages are identified per the coordinate systems shown in Fig. 3(a).

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Fig. 9

Zoomed-in view of the experimentally determined shear stress–strain response and the corresponding Shear modulus

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Fig. 10

Engineering drawing of the APB specimen

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