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Research Papers

Numerical Simulation of Cropping

[+] Author and Article Information
Viggo Tvergaard

Department of Mechanical Engineering,
Technical University of Denmark,
Lyngby2800, Denmark

John W. Hutchinson

School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138-2933

Manuscript received January 9, 2014; final manuscript received February 14, 2014; accepted manuscript posted February 21, 2014; published online March 6, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(7), 071002 (Mar 06, 2014) (9 pages) Paper No: JAM-14-1023; doi: 10.1115/1.4026891 History: Received January 09, 2014; Revised February 14, 2014; Accepted February 21, 2014

Cropping is a cutting process whereby opposing aligned blades create a shearing failure by exerting opposing forces normal to the surfaces of a metal sheet or plate. Building on recent efforts to quantify cropping, this paper formulates a plane strain elastic–plastic model of a plate subject to shearing action by opposing rigid platens. Shear failure at the local level is modeled by a cohesive zone characterized by the peak shear traction and the energy dissipated by shear failure process at the microscopic level. The model reveals the interplay between shear cracking and the extensive plastic shearing accompanying the cutting process. Specifically, it provides insight into the influence of the material’s microscopic shear strength and toughness on the total work of cropping. The computational model does not account for deformation of the cropping tool, friction between sliding surfaces, and material temperature and rate dependence.

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Figures

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Fig. 3

(a) The cohesive traction–displacement relation in shear. (b) The undeformed geometry of the numerical model showing the reference coordinates. The cohesive plane in the undeformed state lies along the x1 axis. The bottom surface along x2 = -a is constrained to undergo frictionless sliding with zero vertical displacement. The upper surface along x2 = a is traction-free. Sliding is suppressed at the contact surfaces between the rigid tools and the metal block.

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Fig. 2

(a) and (b) True shear stress versus log shear strain predicted by the extended Gurson model as dependent on the initial effective void volume fraction f0 and shear damage coefficient kω. The reference stress σR is defined in the text. (c) The normalized specific work of shear separation Γ0/σRD, which is computed for shearing beyond the peak shear stress assuming shearing takes place in a layer of thickness D. For this model, the normalized work is only weakly dependent on the strain hardening exponent N.

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Fig. 1

A cropping shear-off test of Xue et al. [11]. (a) Schematic of the axisymmetric test configuration. The two sections of central plunger are bolted together through the center of the plate whose thickness is h. (b) An experimentally measured nominal shear stress P/(2πRh) versus normalized displacement of the plunger Δ/h for the steel DH 36. For this test: h=3 mm, R=19.05 mm, and the gap width is d≅0.075 mm. Beyond Δ/h = 0.55, the curve drops precipitously. (c) A test interrupted at Δ/h = 0.5 with a section removed from the plate to reveal the details of the shear-off process. (d) A blow up of the region at the corner of the plunger showing that shear cracks have formed, some of which are offset from the corner.

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Fig. 4

(a) The mesh in the undeformed state. The cohesive plane lies along the x1 axis. (b) The deformed mesh for the reference case (case 1) at Δ/h = 0.136.

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Fig. 5

The role of h/RS = π(1-ν2)hτY2/(EΓ0) on the normalized force–displacement relation for cropping for two values of τ∧/τY and with N = 0.185 and τY/E = 0.000785. With all the other dimensional parameters fixed, each set of curves (for a given τ∧/τY) can be interpreted as either the effect of varying Γ0 (with h fixed) or varying h (with Γ0 fixed). Cases 1, 11, 12, 13, 14, 15, and 16.

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Fig. 8

The role of the normalized shear strength in the cohesive zone τ∧/τY on the normalized force–displacement curve with the other dimensionless parameters held fixed. Cases 1, 10, and 11.

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Fig. 9

The role of the strain hardening exponent on the normalized force–displacement curves with the other dimensionless parameters held fixed. Cases 1, 4, 5, 6, and 7.

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Fig. 10

The role of τY/E on the normalized force–displacement curves with the other dimensionless parameters held fixed. As discussed in the text, the specific cropping work with the normalization ΓCROP/Γ0 is only weakly dependent on τY/E with the other dimensionless parameters held fixed. Cases: 1, 2, and 3.

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Fig. 11

A tensile prestress σPS is applied to the plate to produce uniform plastic stretch and then released prior to cropping. The other dimensionless parameters for the prestress cases (cases 17 and 18) are the same as for case 1, the reference case.

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Fig. 12

(a) Undeformed mesh for simulation with a gap d = 2D0. (b) Deformed mesh with gap in the vicinity of the plunger corner for case 19 at Δ/h = 0.108.

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Fig. 13

The role of a gap between the tool corners and the cohesive plane d/h on the normalized force–displacement curves with the other dimensionless parameters held fixed and equal to those of the reference case. Cases: 1, 19, and 20.

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Fig. 6

The dependence of the specific cropping work ΓCROP on the microscopic specific work of fracture Γ0 for plates of the same thickness. Equivalently, the dependence of the specific cropping work ΓCROP on inverse thickness 1/h for plates with fixed material properties. The normalizing quantities denoted by the subscript R refer to reference case 1. The points plotted are cases 1, 11, 12, 13, 14, 15, and 16.

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Fig. 7

Three stages of the cropping process illustrated for the reference case. (i) Onset of cracking when λ first attains 1 in the cohesive zones at the corners of the cropping tool. (ii) At maximum force the cracks at each corner of the tool on opposite surfaces have extended a distance approximately 1/20th the plate thickness. (iii) The point at which the traction plateau is attained in the cohesive zone throughout the central cross section of the plate, i.e., λ≥λ1. As Δ increases beyond this point, localized shear-off is confined to the cohesive zone and the cracks spread toward the center until they meet at Δ/h≅0.16. Case 1.

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