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Research Papers

Rolling at Small Scales

[+] Author and Article Information
Kim L. Nielsen

Associate Professor
Department of Mechanical Engineering,
Technical University of Denmark,
Kongens Lyngby DK-2800, Denmark
e-mail: kin@mek.dtu.dk

Christian F. Niordson

Associate Professor
Department of Mechanical Engineering,
Technical University of Denmark,
Kongens Lyngby DK-2800, Denmark
e-mail: cn@mek.dtu.dk

John W. Hutchinson

Professor
School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138
e-mail: jhutchin@fas.harvard.edu

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received January 25, 2015; final manuscript received July 3, 2015; published online October 27, 2015. Assoc. Editor: Brad L. Kinsey.

J. Manuf. Sci. Eng 138(4), 041004 (Oct 27, 2015) (10 pages) Paper No: MANU-15-1060; doi: 10.1115/1.4031068 History: Received January 25, 2015

The rolling process is widely used in the metal forming industry and has been so for many years. However, the process has attracted renewed interest as it recently has been adapted to very small scales where conventional plasticity theory cannot accurately predict the material response. It is well-established that gradient effects play a role at the micron scale, and the objective of this study is to demonstrate how strain gradient hardening affects the rolling process. Specifically, the paper addresses how the applied roll torque, roll forces, and the contact conditions are modified by strain gradient plasticity. Metals are known to be stronger when large strain gradients appear over a few microns; hence, the forces involved in the rolling process are expected to increase relatively at these smaller scales. In the present numerical analysis, a steady-state modeling technique that enables convergence without dealing with the transient response period is employed. This allows for a comprehensive parameter study. Coulomb friction, including a stick–slip condition, is used as a first approximation. It is found that length scale effects increase both the forces applied to the roll, the roll torque, and thus the power input to the process. The contact traction is also affected, particularly for sheet thicknesses on the order of 10 μm and below. The influences of the length parameter and the friction coefficient are emphasized, and the results are presented for multiple sheet reductions and roll sizes.

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Figures

Grahic Jump Location
Fig. 1

Parameterization of the rolling process under steady-state conditions, with symmetry applied at x2 = 0. Throughout, a·/(ɛ·0H) = 50 and L/H = 10, with the domain discretized by equal sized squared elements of side length; L(e)/H = 20, and unit thickness. The width of the sheet in the out-of-plane direction, b, is not shown.

Grahic Jump Location
Fig. 3

Rolling at large scales (LD/H = 0.05) with low friction (μ = 0.005) and minimum prescribed pull force. (a) Applied torque, (b) applied pull force, (c) applied punch force, and (d) input power determined as P = (Fpull+T/R)a· for T > 0 or P = Fpulla· for T≤0 for various punch displacements, Δ/H (N = 0.1, m = 0.01, σy/E = 0.003,R/H = 100, and stick–slip condition active). The width of the out-of-plane direction is denoted b.

Grahic Jump Location
Fig. 2

Rolling at large scales (LD/H = 0.05) with low friction (μ = 0.005) and minimum prescribed pull force. (a) Applied torque and (b) applied pull force for various punch displacements, Δ/H (N = 0.1, m = 0.01, σy/E = 0.003,R/H = 100, and stick–slip condition active). The width of the out-of-plane direction is denoted b.

Grahic Jump Location
Fig. 11

Rolling at different scales with zero pull force. (a) Applied torque and (b) applied punch force for various punch displacements, Δ/H. Results are shown for three levels of initial yield strain (σy/E = [0.001,0.002,0.003]), with the stick–slip condition active (N = 0.1, m = 0.01, μ = 0.1, and R/H = 100). The width of the out-of-plane direction is denoted b.

Grahic Jump Location
Fig. 4

Effect of friction on rolling at different scales with minimum prescribed pull force. (a) Applied torque, (b) applied pull force, (c) applied punch force, and (d) input power determined as P=(Fpull+T/R)a· for T > 0 or P=Fpulla· for T≤0 for a fixed punch displacement of Δ/H=0.1. Results are shown for various length scales, and with the stick–slip condition active (N = 0.1, m = 0.01, σy/E=0.003, and R/H=100). The width of the out-of-plane direction is denoted b.

Grahic Jump Location
Fig. 5

Contact condition at the roll/sheet interface during rolling: (a) at large scales (LD/H = 0.05) and (b) at small scales (LD/H = 0.50). The normalized friction forces (tangential traction) and normal forces (normal traction) for various friction levels are shown (N = 0.1, m = 0.01, σy/E = 0.003,R/H = 100,Δ/H = 0.1, and stick–slip condition active). Zero pull force is applied.

Grahic Jump Location
Fig. 6

Contact condition at the roll/sheet interface during rolling at different scales. The normalized friction forces (tangential traction) and normal forces (normal traction) for various dissipative length parameters are shown (N = 0.1, m = 0.01, μ = 0.1,σy/E = 0.003,R/H = 100,Δ/H = 0.1, and stick–slip condition active). Zero pull force is applied.

Grahic Jump Location
Fig. 7

Curves of constant effective gradient-enhanced plastic strain, Ep, for a fixed punch displacement of Δ/H = 0.1, and zero pull force. Here, two levels of the dissipative length parameter (a) LD/H = 0.05 and (b) LD/H = 0.50 (N = 0.1, m = 0.01, σy/E = 0.003,R/H = 100,μ = 0.1, and stick–slip condition active) are compared.

Grahic Jump Location
Fig. 8

Effect of roll size for various punch displacements, Δ/H. The applied torque with zero pull force is prescribed. Results are shown for various dissipative length parameters, and with the stick–slip condition active (N = 0.1, m = 0.01, σy = 0.003, and μ = 0.1). Zero pull force is applied. The width of the out-of-plane direction is denoted b.

Grahic Jump Location
Fig. 9

Effect of roll size for a fixed punch displacement of Δ/H = 0.1, showing (a) applied torque and (b) applied punch force. Results are shown for various dissipative length parameters, and with the stick–slip condition active (N = 0.1, m = 0.01, σy/E = 0.003, and μ = 0.1). Zero pull force is applied. The width of the out-of-plane direction is denoted b.

Grahic Jump Location
Fig. 10

Rolling at different scales with zero pull force. (a) Applied torque and (b) applied punch force for various punch displacements, Δ/H. Results are shown for various dissipative length parameters, and for two levels strain hardening (N = [0.1,0.2]) with the stick–slip condition active (m = 0.01, σy/E = 0.003,R/H = 100, and μ = 0.1). The width of the out-of-plane direction is denoted b.

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