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On the size effect for micro-scale structures under the plane bulge test using the modified strain gradient theory

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Abstract

A modified strain gradient theory is proposed based on the nonhomogeneity of polycrystalline metallic materials. Geometrically necessary dislocations are generated on the slip planes as well as on the grain boundary to accommodate deformation with minimum internal stress. Since amount of the geometrically necessary dislocation depends on the deformation shape, specimen size and grain size, it is an important factor for the modified strain gradient theory and the size effect. This new theory differs from the mechanism based strain gradient plasticity in its consideration of the geometrically necessary dislocations on the grain boundary and free surface effect. This theory provides a possible explanation for conflicting size effect: the smaller can be either harder or softer due to the deformation. Using the proposed theory, analysis of the effect of both specimen and grain size under the plane bulge test of polycrystalline materials is performed.

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Abbreviations

a :

radius

α :

an empirical constant usually ranging from 0.2 to 0.5

b S :

Burgers vector

b G :

Burgers vector

d :

grain size

µ:

shear modulus

γ 1, γ 2, γ 3 :

interaction coefficients

L d :

average distance between dislocations

L GS :

dislocation length of the GNDs on the slip planes

L GG :

dislocation length of the GNDs on the grain boundary

ρ :

dislocation density

ρ T :

total dislocation density

ρ S :

density of the statistically stored dislocations

ρ G :

density of the GNDs

ρ GS :

density of the GNDs on the slip planes

ρ GG :

density of the GNDs on the grain boundary

\(\bar m\) :

Taylor factor

σ ref :

reference stress for the uniaxial tension

ɛ :

effective strain

ɛ P :

plastic strain

\(\hat \varepsilon \) :

effective plastic strain

N :

work hardening exponent (0 ≤ N < 1)

l :

material characteristic length

η E :

effective strain gradients

η S :

strain gradients caused by the slip plane

η G :

strain gradients caused by grain boundary

\(\bar r\) :

Nye factor

R :

radius of curvature

ϑ :

angle

h :

deflected height of the film

L 0 :

initial length

T 0 :

initial thickness

W 0 :

initial width of the specimens

L :

length of the specimens after deformation

T :

thickness of the specimens after deformation

W :

thickness of the specimens after deformation

P :

applied pressure

V :

plastic volume of specimen

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Authors and Affiliations

  1. Department of Materials Science and Engineering, MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Hunkee Lee

  2. School of Mechanical Engineering, POSTECH, San 31, Hyoja-dong, Nam-gu, Pohang, Gyungbuk, South Korea, 790-784

    Bongbu Jung, Dongseob Kim & Hyunchul Park

Authors
  1. Hunkee Lee
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  2. Bongbu Jung
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  3. Dongseob Kim
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  4. Hyunchul Park
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Correspondence to Hyunchul Park.

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Lee, H., Jung, B., Kim, D. et al. On the size effect for micro-scale structures under the plane bulge test using the modified strain gradient theory. Int. J. Precis. Eng. Manuf. 12, 865–870 (2011). https://doi.org/10.1007/s12541-011-0115-7

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