Archive of Applied Mechanics

, Volume 88, Issue 4, pp 587–612 | Cite as

On the use of the extended finite element and incremental methods in brittle fracture assessment of key-hole notched polystyrene specimens under mixed mode I/II loading with negative mode I contributions

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Abstract

The aim of the present study is to assess the suitability of the extended finite element method (XFEM) combined with the cohesive zone model (CZM) and also the incremental method together with the maximum tangential stress (MTS) criterion in predicting the fracture load and crack trajectory of key-hole notched brittle components subjected to mixed mode I/II loading with negative mode I contributions. For this purpose, a total number of 63 fracture test results, reported recently in the literature on the key-hole notched Brazilian disk (Key-BD) specimens made of the general-purpose polystyrene (GPPS) under mixed mode I/II loading with negative mode I contributions, are first collected. Then, the experimentally obtained fracture loads of the tested GPPS specimens are theoretically predicted by means of XFEM combined with CZM. Additionally, the crack trajectory in the tested Key-BD specimens is predicted by using both XFEM combined with CZM and the incremental method combined with MTS criterion. Finally, it is shown that both the fracture load and the crack trajectory could successfully be predicted by means of the two proposed methods for different notch geometries.

Keywords

Brittle fracture Cohesive zone model (CZM) Extended finite element method (XFEM) Key-hole notch Negative mode I 

List of symbols

ASED

Averaged strain energy density

ASED-EFC

Averaged strain energy density based on the equivalent factor concept

\(b_{i}\)

Gradient vector of the shape function associated with node i

CTSN

Compact-tension-shear-notched

CZM

Cohesive zone model

D

Fourth-order elastic moduli tensor

E

Young’s modulus

ES

The element size applied to the notch border

\(f(\,)\)

Softening function

\(G_\mathrm{f}\)

Specific fracture energy

FIA

Fracture initiation angle

FVSD

Flattened V-notched semi-disk

GPPS

General-purpose polystyrene

\(H(\,)\)

Heaviside function

Key-MS

Key-hole notch mean stress

Key-MTS

Key-hole notch maximum tangential stress

\(K_{\mathrm{I}}\)

Mode I stress intensity factor

\(K_{\mathrm{II}}\)

Mode II stress intensity factor

\(K_{\mathrm{Ic}}\)

Plane strain fracture toughness

Key-BD

Key-hole notched Brazilian disk

LEFM

Linear elastic fracture mechanics

LCC

Load-carrying capacity

\(l_\mathrm{ch}\)

Characteristic length

\(L_{1}\)

Total slit length in the Key-BD specimen

\(L_{2}\)

Diameter of the Key-BD specimen

MS

Mean stress

MTS

Maximum tangential stress

n

Unitary vector normal to the maximum principal stress

NFM

Notch fracture mechanics

\(N_{i}(\,)\)

Shape function associated with node i

PS

Point stress

RNL

Relative notch length

SED

Strain energy density

SIF

Stress intensity factor

t

Traction vector

T

Cohesive traction

u()

Displacements field

\(u_{i}\)

Nodal displacements of node i

VSC

V-notched stepped cottage

w

Crack opening vector

\({\tilde{w}} \)

Equivalent crack opening

XFEM

Extended finite element method

\(\beta \)

Loading angle in the Key-BD specimen

\({\beta }_{\mathrm{II}}\)

Loading angle corresponding to pure mode \(\mathrm{I}\mathrm{I}\) loading

\(\delta \)

Virtual crack opening displacement

\(\nu \)

Poisson’s ratio

\(\rho \)

Notch tip radius

\({\sigma }_{\mathrm{u}} \)

Ultimate tensile strength

\({\sigma }_{\vartheta \vartheta } \)

Tangential stress

\(\vartheta _{{0,}{\mathrm{Exp.}}}\)

Fracture initiation angle obtained from the experiment

\(\vartheta _{{0,}{\mathrm{Key-MTS}}}\)

Fracture initiation angle obtained from the key-hole notch maximum tangential stress criterion

\(\vartheta _{{0,}{\mathrm{XFEM}}}\)

Fracture initiation angle obtained from the extended finite element method

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • H. R. Majidi
    • 1
  • M. R. Ayatollahi
    • 1
  • A. R. Torabi
    • 2
  1. 1.Fatigue and Fracture Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical EngineeringIran University of Science and TechnologyNarmak, TehranIran
  2. 2.Fracture Research Laboratory, Faculty of New Sciences and TechnologiesUniversity of TehranTehranIran

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