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Connecting the essential work of fracture, stress intensity factor, Hill’s criterion in mixed mode I/II loading

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Abstract

Ductile sheet structures are frequently subjected to mixed mode loading, resulting that the structure is under the influence of a mixed mode stress field. Instances of interest are when stable crack growth occurs and when the crack-tip is propagating in this complex mixed-mode condition, prior to final fracture. Purposely designed apparatus was built to test thin-sheets of steel (Grade: DX51D) under mixed-mode I/II. These tests, under plane stress conditions, also investigated the effect of thickness on the specific essential work of fracture or the fracture toughness of the material under quasi-static cracking conditions. The fracture toughness is evaluated under incremental mixed-mode loading conditions. The direction of the propagating crack path and fracture type were observed and discussed as the loading mixity was varied. Whilst the specific essential work of fracture or fracture toughness was obtained using the energy approach, the theoretical analysis of the fracture type and direction of crack path were based on the crack tip stresses and fracture criterions of maximum hoop stress and maximum shear stress along with the utilisation of Hill’s theory. For mixed-mode I/II loading, the variation in the fracture toughness contributions ratios are evaluated and used predicatively using the established energy criterion approach to the crack tip stress intensity approach. The comparison between the theoretical directions of the crack path, failure mode propagation are in good agreement with those obtained from experimental testing indicating the definite link between both approaches.

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Abbreviations

\(a\) :

Crack length (m)

\(A\) :

Surface area of the crack \((\hbox {m}^{2})\)

\(d\) :

Constant

\(E\) :

Young’s Modulus \((\hbox {N/m}^{2})\)

\(H\) :

Height (m)

\(K\) :

Stress Intensity Factor \((\hbox {Pa}\sqrt{m})\)

\(L\) :

Ligament length (m)

\(R\) :

Fracture Toughness (specific essential work of fracture) (N/m)

\(t\) :

Thickness (m)

\(u\) :

Displacement (m)/Length (m)

\(U\) :

Work done/Energy (J)

\(V\) :

Volume of the body \((\hbox {m}^{3})\)

\(V^{e}\) :

Volume of the body undergoing elastic deformation \((\hbox {m}^{3})\)

\(V^{P}\) :

Volume of the body undergoing plastic flow \((\hbox {m}^{3})\)

\(w\) :

Width (m)

\(W\) :

Work done (J)

\(W^{e}\) :

Elastic strain energy density (J)

\(W^{P}\) :

Plastic work done \(= \smallint \bar{\sigma } \,d\, \bar{\epsilon }\) (J)

x/y/z:

Coordinates

\(X\) :

Load (N)

\(W\) :

Elastic strain energy density \((\hbox {J/m}^{3})\)

\(W^{P}\) :

Plastic work done per unit volume \((\hbox {J/m}^{3})\)

\(\alpha \) :

Constant

\(\beta \) :

Constant

\(\upvarepsilon \) :

Strain

\(\phi \) :

Direction of relative motion

\(\varLambda \) :

Elastic strain energy

\(\eta \) :

Loading angle for Mode I/II

\(\sigma \) :

Stress \((\hbox {N/m}^{2})\)

\(\tau \) :

Shear stress \((\hbox {N/m}^{2})\)

\(\theta \) :

Direction of stress

\(\varGamma \) :

Plastic strain energy

DEN:

Double End Notched

MHSC:

Maximum Hoop Stress Criterion

MSSC:

Maximum Shear Stress Criterion

SMVS:

Stereo machine vision system

Avg:

Average

c:

Critical

comp:

Compression

I:

Mode I component

II:

Mode 2 component

I/II:

Mixed-mode I/II component

max:

Maximum

min:

Minimum

m/c:

Machine

rel:

Relative

p:

Plastic

T:

Total

ten:

Tension

y:

Yield

*:

Mode I component

**:

Mode II component

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, University of Malta, Msida, MSD , 2080, Malta

    Claire De Marco Muscat-Fenech & Stephen Ciappara

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  1. Claire De Marco Muscat-Fenech
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  2. Stephen Ciappara
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Correspondence to Claire De Marco Muscat-Fenech.

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De Marco Muscat-Fenech, C., Ciappara, S. Connecting the essential work of fracture, stress intensity factor, Hill’s criterion in mixed mode I/II loading. Int J Fract 183, 187–202 (2013). https://doi.org/10.1007/s10704-013-9886-4

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  • DOI: https://doi.org/10.1007/s10704-013-9886-4

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