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
References
Abaqus (2012) Ābaqus ver 6.12 User’s manual. Dassault Systemes K. K., Kaigan Minato-Ku
Al-Ani A, Hancock JW (1991) J-dominance of short cracks in bending and tension. J Mech Phys Solids 39:23–43
ASTM-E1820–11 (2011) Standard test method for measurement of fracture toughness. American Society For Testing And Materials
Betegon C, Hancock JW (1991) Two-parameter characterization of elastic-plastic crack-tip fields. J Appl Mech 58:104–110
Bilby BA, Cardew BA, Goldthorpe MR, Howard IC (eds) (1986) A finite element investigation of the effect of specimen geometry on the field of stress and strain at the tip of a stationary cracks. Institution Of Mechanical Engineers, London
Brocks W, Olschewski J (1986) On J-dominance of crack tip fields in largely 3D structures. Int J Solids Struct 22:693–708
BS7910 (2005) Guidance on methods for assessing the acceptability of flaws in metallic structures. British Standard Institution, London
Dodds RH, Anderson TL, Kirk MT (1991) A framework to correlate a/w effects on elastic-plastic fracture toughness (Jc). Int J Fract 48:1–22
Du ZZ, Hancock JW (1991) The effect of non-singular stresses on crack-tip constraint. J Mech Phys 39:555–567
Graba M (2017) On the parameters of geometric constraints for cracked plates under tension—three-dimensional problems. Int J Appl Mech Eng 22:901–919
Guo WL (1993a) Elastoplastic 3-dimensional crack border field. I. Singular structure of the field. Eng Fract Mech 46:93–104
Guo WL (1993b) Elastoplastic 3-dimensional crack border field. II. Asymptotic solution for the field. Eng Fract Mech 46:105–113
Guo WL (1995) Elastoplastic 3-dimensional crack border field. III Fracture parameters. Eng Fract Mech 51:51–71
Hebel J, Hohe J, Friedmann V, Siegele D (2007) Experimental and numerical analysis of in-plane and out-of-plane crack tip constraint characterization by secondary fracture parameters. Int J Fract 146:173–188
Henry BS, Luxmoore AR (1997) The stress triaxiality constraint and the Q-value as a ductile parameter. Eng Fract Mech 57:375–390
Hom CL, Mcmeeking RM (1990) Large crack tip opening in thin elastic-plastic sheets. Int J Fract 45:103–122
Hutchinson JW (1968) Singular behaviour at the end of a tensile crack tip in hardening material. J Mech Phys Solids 16:13–31
Kim Y, Zhu XK, Chao YJ (2001) Quantification of constraint on elastic-plastic 3D crack front by the J-A2 three term solution. Eng Fract Mech 68:895–914
Kim Y, Chao YJ, Zhu XK (2003) Effect of specimēn size and crack depth on 3D crack-front constraint for senb specimens. Int J Solids Struct 40:6267–6284
Kuna M (2013) Finite elements in fracture mechanics: theory-numerics-applications. Springer, Wiesbaden
Larsson SG, Carlsson AJ (1973) Influence of non-singular stress terms and specimen geometry on small scale yielding at crack tips in elastic plastic material. J Mech Phys Solids 21:263–278
Leong KH, Yusof F, Latiff RHA (2021) Automatic crack tip meshing approach for constraint-based fracture mechanics application. J Fail Anal Prev 21:806–821
McMeeking RM, Parks DM (1979) On criteria for j-dominance of crack tip fields in large scale yielding. Astm Stp 668. American Society For Testing And Materials, Philadelphia
Meshii T, Tanaka T (2010) Experimental T33-stress formulation of test specimen thickness effect on fracture toughness in the transition temperature region. Eng Fract Mech 77:867–877
Moran B, Shih CF (1987) A general treatment of crack tip contour integrals. Int J Fract 35:295–310
Mostafavi M, Smith DJ, Pavier MJ (2010) Reduction of measured toughness due to out-of-plane constraint in ductile fracture of aluminum alloy specimens. Fatigue Fract Eng Mater Struct 33:724–739
Nakamura T, Parks DM (1990) Three-dimensional crack front fields in a thin ductile plate. J Mech Phys Solids 38:787–812
Nevalainen M, Dodds RH (1995) Numerical investigation of 3-D constraint effects on brittle fracture in SE(B) and C(T) specimens. Int J Fract 74:131–161
Newman JC, Bigelow CA, Shivakumar KN (1993) Three-dimensional elastic-plastic finite-element analyses of constraint variations in cracked bodies. Eng Fract Mech 46:1–13
O’Dowd NP (1995) Application of two parameter approaches in elastic plastic fracture mechanics. Eng Fract Mech 52:445–465
O’Dowd NP, Shih CF (1991) Family of crack-tip fields characterized by a triaxiality parameter—I. Structure of fields. J Mech Phys Solids 39:989–1015
O’Dowd NP, Shih CF (1992) Family of crack tip fields characterised by a triaxiality parameter: Part II—fracture applications. J Mech Phys Solids 40:939–963
Prandtl L (1920) Ueber Die Haerte Plastischer Koerper. Goettinger Nachr. Math.-Phys KI 74–85
R6 (2001) Assessment of the integrity of structures containing defects. Rev 4 ed. British Energy Generation Ltd, Gloucester
Rice JR (1968) Mathematical analysis in the mechanics of fracture. In: Liebowitz H (ed) Fracture: An advanced Treatise. Academic Press, New York
Rice JR (1974) Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 22:17–26
Rice JR (1982) Elastic-plastic crack growth. In: Hopkins HG, Sewell MJ (eds) The Rodney Hill 60th Anniversary Volume. Pergamon Press, Oxford
Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power law hardening material. J Mech Phys Solids 16:1–12
Rice JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 17:201–217
Ritchie RO, Knott JF, Rice JR (1973) On the relationship between critical tensile stress and fracture toughness in mild steel. J Mech Phys Solids 21:395–410
Shih CF (1983) Tables of Hutchinson-Rice-Rosengren singular field quantities. Brown University, Providence
Shih CF, German MD (1981) Requirements for a one parameter characterization of crack tip fields by the HRR singularity. Int J Fract 17:27–43
Shih CF, Moran B, Nakamura T (1986) Energy release rate along a three-dimensional crack front in a thermally stressed body. Int J Fract 30:79–102
Shlyannikov VN, Boychenko NV, Tumanov AV, Fernandez-Canteli A (2014) The elastic and plastic constraint parameters for three-dimensional problems. Eng Fract Mech 127:83–96
Sorem WA, Dodds RH, Rolfe ST (1989) An analytical and experimental comparison of rectangular and square crack-tip opening displacement fracture specimens on an A36. ASTM STP 995. American Society For Testing And Materials, Philadelphia
Wellman GW, Dodds RH, Sorem WS, Rolfe ST (1985) Three-dimensional elastic-plastic finite element analysis of three-point bend specimens. ASTM STP 688. American Society Of Testing And Materials, Philadelphia
Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114
Yang J, Wang G, Xuan F, Tu S (2013) Unified characterisation of in-plane and out-of-plane constraint based on crack-tip equivalent plastic strain. Fatigue Fract Eng Mater Struct 36:504–514
Yang S, Chao YJ, Sutton MA (1993) Higher order asymptotic crack tip fields in a power law hardening material. Eng Fract Mech 45:1–20
Yuan H, Brocks W (1998) Quantification of constraint effects in elastic-plastic crack front fields. J Mech Phys Solids 46:219–241
Yusof F (2019) Three-dimensional assessments of crack tip constraint. Theoret Appl Fract Mech 101:1–16
Yusof F, Hancock JW (2005) In-plane and out-of-plane constraint effects in three-dimensional elastic perfectly-plastic crack tip field. In: 11th international conference on fracture. Turin
Yusof F, Leong KH (2019) Elastic-plastic J-Tz dominance in bending and tension loadings. Int J Struct Integrity 10:644–659
Zhao JH, Guo WL, She CM (2008) Three-parameter approach for elastic-plastic fracture of the semi-elliptical surface crack under tension. Int J Mech Sci 50:1168–1182
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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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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DOI: https://doi.org/10.1007/s10704-021-00571-6