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A Criterion for ductile fracture in sheets under biaxial loading

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Abstract

A criterion for ductile fracture is developed based on the statistical process of shear joining of voids and on the assumption that the voids responsible for fracture have experienced considerable growth prior to this stage of shearing. From the knowledge of uniaxial flow properties and fracture strain measurement, this model is capable of predicting the strain at fracture for other strain states. The predicted data are in good agreement with experiments. Although this model assumes spherical inclusions, some quantitative estimates for elongated inclusions can also be made.

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Abbreviations

σ 1,σ 2,σ 1 :

Maximum, intermediate and minimum principal stresses;σ 1 andσ 2 act in the plane of the sheet andσ 3 = 0σ m Mean normal (or triaxial) stress = (σ 1 +σ 2 +σ 3)/3

σ p p :

Mean normal stress in a plane perpendicular to elongated inclusions

τ max :

Maximum shear stress = (σ 1 -σ 3)/2

•τ :

Effective shear stress

dλ :

Plastic modulus

ε 1,ε 2,ε 3 :

Respectively, the algebraically maximum, intermediate and minimum plastic strains (ε 1 andε 2 are in the plane of the sheet)

ε 1f,ε 2f,ε 3f :

The same strains at fracture

ε :

Effective plastic strain rate

α :

Constant planar stress-ratio (σ 2/σ 1)

ρ,ρ u,ρ o :

Constant planar strain ratio (ε 2/ε 1. The subscriptu refers to uniaxial tension ando refers to any other strain-state between uniaxial and plane strain tension

χ,M 1,M 2 :

Proportionality constants

R :

Normal anisotropy parameter (assuming planar isotropy), defined by the ratio of the width to thickness strains in a tensile test

n, C :

Respectively the exponent and the constant in the parabolic hardening law:σ = Cεn

Λ,m :

Respectively the intervoid spacing and a related constant

f :

Volume fraction of inclusions

S :

Rate of increase in the surface area of voids

A,A :

Midsectional area of voids from spherical inclusions and its growth rate, respectively

R 0 :

Original radius of a spherical inclusion of average size

R :

Radial growth rate of voids from spherical inclusions

l, d, A c :

Respectively, the length, diameter, and midsectional area of a cylindrical inclusion with spherical ends

A l,A d :

Respectively, the growth rates of the cylindrical and spherical midsectional areas of a void from cylindrical inclusion

g :

Geometrical factor related to the directional growth rates of a void from cylindrical inclusion (≃1.5)

P :

Probability of shear linking of voids

K cr :

A material constant related to the critical probability of shear linking

C*,C*1,D*:

Constants related to critical values at fracture

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Authors and Affiliations

  1. Physics Department, General Motors Research Laboratories, 48090, Warren, MI

    Amit K. Ghosh (Senior Research Metallurgist)

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  1. Amit K. Ghosh
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Ghosh, A.K. A Criterion for ductile fracture in sheets under biaxial loading. Metall Trans A 7, 523–533 (1976). https://doi.org/10.1007/BF02643968

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