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Three-dimensional crack tip constraint of shallow cracks in tension and bending

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Abstract

Two-dimensional plane strain approaches in fracture mechanics have been used to characterize crack tip constraint of cracked geometries from deep to shallow cracks but neglected out-of-plane crack tip constraint effect. To address the effect of thickness and crack length in three-dimensional crack tip constraint, fully constrained geometries of notched bend bars and unconstrained geometries of center cracked tension panels of deep to shallow cracks in non-hardening and hardening elastic–plastic crack tip fields have been examined. From the results, it is found that thickness affects the crack tip constraint of deep and shallow cracks by changing the shape of the plastic zones and hence the normal stresses at the crack tip. The reduction of crack length from deep to shallow cracks in fully constrained and unconstrained crack tip fields by maintaining the ratio of \(B\)/(\(W\)–\(a\)) through an increase in thickness caused the normal stresses at the crack tip to increase marginally and led to a reduction of the toughness of the shallow cracked geometries. The change in the toughness due to the change in crack length and thickness can be characterized through a \(J\)–\({\Delta }\sigma\) technique which is based on a crack tip constant stress sector difference fields approach.

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Abbreviations

\(J\) :

\(J\)-Integral

\(\mu\) :

Non-dimensional classification parameter

\(T\) :

Non-singular \(T\)-stress

\(K\)–\(T\) :

Two parameter fracture mechanics—small scale yielding in-plane strain crack configurations

\(J\)–\(Q^{2D}\) :

Two parameter fracture mechanics—moderate large scale yielding in-plane strain crack configurations

\(J\)–\(A_{2}\) :

Two parameter fracture mechanics—for higher-order terms in-plane strain crack configurations

\(Q^{2D}\) :

Non-singular term in \(HRR\) field

\(HRR\) :

Hutchinson-Rice-Rosengren

\(\sigma_{ij}^{2D}\) :

Cartesian stresses in plane strain crack solution

\(\sigma_{ij}^{HRR}\) :

Cartesian stresses for HRR fields

\(\sigma_{o}\) :

Initial yield stress based on von Mises in uniaxial tension

\(\delta_{ij}\) :

Kronecker delta

\(\sigma_{\theta \theta }\) :

Hoop stress

\(r\) :

The radial distance ahead of a crack tip

\(\theta\) :

Angle around crack tip in degrees

\(\sigma_{\theta \theta }^{2D}\) :

Hoop stress from a two-dimensional analysis

\(\sigma_{\theta \theta }^{HRR}\) :

Hoop stress from a HRR field

\(x_{3}\)/\(B\) :

Length measured normal from the mid-plane of a cracked plane normalized by the thickness of a cracked specimen

\(J\)–\(Tz\) :

Two parameter fracture mechanics—a measure of out-of-plane constraint

\(Q^{*}\) :

Out-of-plane constraint measure based on \(J\)–\(Tz\)

\(\sigma_{22}\) :

Crack opening stress

\(J\)–\(Tz\)–\(Q^{2D}\) :

Three-parameter fracture mechanics—a measure of out-of-plane and in-plane constraint

\(z\) :

Length measured normal from the free surface of a cracked plane

\(\sigma_{ij}^{op} /\sigma_{o}\) :

Cartesian stresses in the out-of-plane normalized by the initial yield stress

\(\left( {\frac{{\sigma_{ij} }}{{\sigma_{o} }}} \right)_{T,Q}^{pl.\varepsilon }\) :

Cartesian stresses from a two-dimensional plane strain model dependent on the in-plane constraint

\(\left( {\frac{{\sigma_{ij} }}{{\sigma_{o} }}} \right)^{pl.\sigma }\) :

Cartesian stresses from a two-dimensional plane stress model

\(\gamma_{{\left( {r,n} \right)}}\) :

Out-of-plane constraint sensitivity dependent on the distance ahead of the crack (\(\theta \) = 0°) and strain hardening exponent, \(n\)

\(J_{loc}\) :

Local \(J\)-integral along a crack front tip

\(n\) :

Strain hardening exponent

\(\sigma_{1}\)/\(\sigma_{y}\) :

A measure of constraint-based on principal stress contour around the crack tip

\(T_{33}\) :

An out-of-plane measure of constraint based on \(T\)

\(J\)–\(\sqrt \phi\) :

Two-parameter fracture mechanics—a measure of out-of-plane constraint based on

\(J\)–\(\sqrt {A_{p} }\) :

Two-parameter fracture mechanics—a measure of out-of-plane constraint based on an equivalent plastic strain

\(H\)/\(W\) :

Half-length normalized by ligament width of a cracked model

\(a\)/\(W\) :

Crack length normalized by ligament width of a cracked model

\(B\)/(\(W\)–\(a\)):

Thickness normalized by the difference of ligament width to crack length

SENB:

Single edge notched bend bar

CCP:

Center-cracked panel

\(x_{3}\) :

Length measured normal from the mid-plane of a cracked plane

\(r_{e}\) :

The radius of the element at the crack tip

\(r_{sa}\) :

The radius of a semi-annular crack tip region

\(\varepsilon\) :

Strain

\(\sigma\) :

Stress

\(\alpha\) :

Material’s constant in Ramberg–Osgood materials response

\(J_{2}\) :

Von Mises yield criteria

\(\varepsilon_{ij}\) :

Infinitesimal strain tensor

\(\sigma_{ij}\) :

Cartesian component of a stress tensor

\(\sigma_{e}\) :

Equivalent von Mises stress

\(S_{ij}\) :

Deviatoric stress

\(E\) :

Modulus of elasticity

\(c\) :

Difference of cracked and uncracked ligament

\(Q^{3D}\) :

Out-of-plane constraint

\(\sigma_{\theta \theta }^{{3D\left( {FE} \right)}}\) :

Hoop stress from a three-dimensional finite element model

\(\infty\) :

Infinity

\(\sigma_{critical}\) :

Critical stress

\(\sigma_{m}\) :

Mean stress

\(\frac{{{\Delta }\sigma }}{{\sigma_{o} }}^{3D}\) :

A measure of in-plane and out-of-plane crack tip constraint

\(\frac{{{\Delta }\sigma }}{{\sigma_{o} }}^{ip}\) :

A measure of in-plane constraint

\(\frac{{{\Delta }\sigma }}{{\sigma_{o} }}^{op}\) :

A measure of out-of-plane constraint

\(\left( {\frac{{\sigma_{\theta \theta }^{op} }}{{\sigma_{o} }}} \right)\) :

Hoop stress out-of-plane constraint

\(\left( {\frac{{\sigma_{\theta \theta } }}{{\sigma_{o} }}} \right)^{ref}\) :

Reference hoop stress

\(\left( {\frac{{\sigma_{\theta \theta } }}{{\sigma_{o} }}} \right)_{{Q^{2D} }}^{HRR/SSY pl.\varepsilon }\) :

Hoop stress for a HRR or small-scale yielding from a two-dimensional plane strain model

\(J\)–\({\Delta }\sigma\) :

Two parameter fracture mechanics—in-plane and out-of-plane constraint based on a crack tip constant stress sector difference fields approach

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Acknowledgements

Leong Karh Heng is pleased to be able to acknowledge the support of a Malaysia’s Ministry of Higher Education (MOHE) grant (FRGS 2016/F1123) and thanks are due to Dassault Systemes K. K. Japan for access to ABAQUS available in the Universiti Sains Malaysia.

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  1. School of Mechanical Engineering, Universiti Sains Malaysia, 14300, Nibong Tebal, Penang, Malaysia

    K. H. Leong & F. Yusof

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  1. K. H. Leong
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Correspondence to F. Yusof.

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Leong, K.H., Yusof, F. Three-dimensional crack tip constraint of shallow cracks in tension and bending. Int J Fract 231, 169–187 (2021). https://doi.org/10.1007/s10704-021-00571-6

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