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Theoretical study of void closure in nonlinear plastic materials

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Abstract

Void closing from a spherical shape to a crack is investigated quantitatively in the present study. The constitutive relation of the void-free matrix is assumed to obey the Norton power law. A representative volume element (RVE) which includes matrix and void is employed and a Rayleigh-Ritz procedure is developed to study the deformation-rates of a spherical void and a penny-shaped crack. Based on an approximate interpolation scheme, an analytical model for void closure in nonlinear plastic materials is established. It is found that the local plastic flows of the matrix material are the main mechanism of void deformation. It is also shown that the relative void volume during the deformation depends on the Norton exponent, on the far-field stress triaxiality, as well as on the far-field effective strain. The predictions of void closure using the present model are compared with the corresponding results in the literature, showing good agreement. The model for void closure provides a novel way for process design and optimization in terms of elimination of voids in billets because the model for void closure can easily be applied in the CAE analysis.

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Abbreviations

A,A (i) k :

coefficients in stream function

Ė :

macroscopic strain-rate tensor

Ė 0, v 0 :

strain-rate and velocity in the absence of the void

n :

Norton exponent

n :

unit normal vector

r, θ, φ :

spherical coordinates

sign:

signum function

S R, S :

surface of the RVE, surface of the void

v,\( \tilde \upsilon \) :

local velocity, additional velocity

V R, V M :

volume of the RVE, volume of the matrix material in the RVE

V, V 0 :

current void volume, initial void volume

σ,σ′, σ e :

local stress tensor, stress deviator and effective stress

σ 0, \( \dot \varepsilon \) 0 :

reference stress, reference strain-rate

\( \dot \varepsilon ,\dot \tilde \varepsilon ,\dot \varepsilon _e \) :

local strain-rate tensor, additional strain-rate tensor and effective strain-rate

Σ,Σ′ :

macroscopic stress tensor and stress deviator

Σm,Σe :

macroscopic mean stress and effective stress

ζ :

stream function

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Authors and Affiliations

  1. National Die and Mold CAD Engineering Research Center, Shanghai Jiao Tong University, Shanghai, 200030, P. R. China

    Xiao-xun Zhang  (张效迅) & Zhen-shan Cui  (崔振山)

Authors
  1. Xiao-xun Zhang  (张效迅)
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  2. Zhen-shan Cui  (崔振山)
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Correspondence to Zhen-shan Cui  (崔振山).

Additional information

(Communicated by Zhu-ping HUANG)

Project supported by the National Basic Research Program of China (973 Program) (No. 2006CB705401)

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Zhang, Xx., Cui, Zs. Theoretical study of void closure in nonlinear plastic materials. Appl. Math. Mech.-Engl. Ed. 30, 631–642 (2009). https://doi.org/10.1007/s10483-009-0509-7

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  • DOI: https://doi.org/10.1007/s10483-009-0509-7

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