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Theoretical approach of characterizing the crack-tip constraint effects associated with material’s fracture toughness

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Abstract

The conversion problem of plane strain fracture toughness (\(K_\mathrm{IC}\)) which is necessarily measured according to ASTM standards, to lower constraint applications generally existing in engineering structures, has led to extensive material and labor costs. The present paper explores and quantifies the crack-tip constraint effects by the crack-tip plastic zone to serve the prediction of material’s fracture toughness. Firstly, the approximate three-dimensional crack front displacement fields are obtained by using the variable separation method. The three-dimensional stress field is then used to predict the shape and size of the crack-tip plastic zone. Secondly, the two-dimensional in-plane and the three-dimensional out-of-plane geometric constraint effects are quantified separately, and two constraint factors, i.e., \(\alpha _\mathrm{in}\) and \(\alpha _\mathrm{out}\) are proposed. A series of configurations of the two-dimensional and three-dimensional crack-tip plastic zones that vary with the specimen’s geometric size (plate width, thickness, etc.) are presented, which will facilitate a better understanding of the crack-tip constraint effect. Finally, the present method is applied to elucidate the significant “thickness effect” of the X70 pipeline steel’s fracture toughness by using the out-of-plane constraint factor \(\alpha _\mathrm{out}\). The corresponding results are compared with the experimentally measured and FEM-predicted fracture toughness \(K_\mathrm{C}\). It is concluded that the parameters \(\alpha _\mathrm{out}\) and the fracture toughness \(K_\mathrm{C}\) can be correlated with each other, which will be beneficial to the prediction of material’s fracture toughness and the avoidance of experimental costs.

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Abbreviations

2D:

Two-dimensional

3D:

Three-dimensional

a :

Crack length

\(A_{0}\) :

Area of crack-tip plastic zone with infinite plate width in-plane stress

\(A_{2}\) :

Secondary fracture parameter in \(J{-}A_{2}\) concept

\(A_{n}\) :

Area of crack-tip plastic zone in-plane strain

\(A_{\mathrm{p}}\) :

A unified constraint parameter based on the equivalent plastic train

\(A_{y}\) :

Area of crack-tip plastic zone with finite width—plane stress state

B :

Plate thickness

CT:

Compact tension

E :

Young’s modulus

F :

The load applied to the three-point bending specimen

FEM:

Finite element method

\(F_{Q}\) :

The critical load of three-point bending test at fracture

f(S):

The geometric correction factor of \(K_{\mathrm{I}}\)

\(G_\mathrm{C}\) :

Critical energy release rate at fracture

J :

J-integral

\(J_\mathrm{C}\) :

Fracture toughness characterized by J-integral

h :

Half-thickness of the 3D plate

\(K_\mathrm{C}\) :

Fracture toughness

\(K_\mathrm{I}\) :

Stress intensity factor of mode I crack

\(K_\mathrm{IC}\) :

Plane strain fracture toughness

L :

Plate length

Q :

A constraint parameter based on elastic–plastic finite element method calculations

\(r_{0}\) :

Crack-tip elastic–plastic boundary for infinite plate width—plane stress state

\(r_{K}\) :

Radius of the elastic stress intensity factor \(K_\mathrm{I}\) dominated zone

\(r_{n}\) :

The crack-tip elastic–plastic boundary line—plane strain state

\(r_{y}\) :

Crack-tip elastic–plastic boundary

\(r,\theta , z\) :

Cylindrical coordinates

S :

The ratio of crack length to width of plate

\(S_{L}\) :

The span length of SENB specimen

SENB:

Single-edge notched bend

SIF:

Stress intensity factor

SSY:

Small-scale yielding

T :

T-stress

\(T_{z}\) :

Out-of-plane stress constraint factor

\(T_{33}\) :

Out-of-plane component of T-stress

u, v, w :

The 3D crack front displacement field

\(u_{n}, v_{n}\) :

Crack front x- and y-direction displacement—plane strain state

\(V_{y}\) :

Volume of 3D crack-tip plastic zone

\(w_{s}\) :

Crack front z-direction displacement—plane stress state

W :

Plate width

x, y, z :

Rectangular coordinates

\(\alpha _\mathrm{in}\) :

In-plane constraint factor

\(\alpha _\mathrm{out}\) :

Out-of-plane constraint factor

\(\lambda \), \(\mu \) :

Material’s Lame’s constants

\(\eta (z), \zeta (z)\) :

The undetermined functions of 3D crack front displacement

\(\upsilon \) :

Poisson’s ratio

\(\sigma \) :

Far-field tensile load

\(\sigma _{i}\) :

Principal stresses

\(\sigma _{s}\) :

Yield strength of material

\(\sigma _{e}\) :

Huber–Mises–Hencky stress

\(\sigma _{x}, \sigma _{y}, \sigma _{z}\) :

3D crack front normal stress components

\(\tau _{xy}, \tau _{xz}, \tau _{yz}\) :

3D crack front shear stress components

\(\varepsilon _{x},\varepsilon _{y}, \varepsilon _{z}\) :

3D crack front normal strain components

\(\varepsilon _{xn},\varepsilon _{yn}\) :

The crack front strain components—plane strain state

\(\varepsilon _{\mathrm{p}}\) :

The equivalent plastic strain y

\(\gamma _{xy}, \gamma _{xz}, \gamma _{yz}\) :

The 3D crack front shear strain components

\(\varPhi \) :

A constraint parameter defined by the area of plastic zone

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos.11772245, 11472205), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2014K10-16) and the Fundamental Research Funds for the Central Universities in China.

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Authors and Affiliations

  1. State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, 710049, Xi’an, China

    Junnan Lv, Li Yu & Qun Li

  2. Nuclear Power Institute of China, Chengdu, 610200, China

    Li Yu

  3. School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Wei Du

  4. CNPC Tubular Goods Research Institute, Xi’an, 710077, China

    Wei Du

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  1. Junnan Lv
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Lv, J., Yu, L., Du, W. et al. Theoretical approach of characterizing the crack-tip constraint effects associated with material’s fracture toughness. Arch Appl Mech 88, 1637–1656 (2018). https://doi.org/10.1007/s00419-018-1392-8

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