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Mixed-mode I/II fracture criterion for crack initiation assessment of composite materials

  • Mahdi Fakoor
  • Hannaneh Manafi Farid
Original Paper
  • 3 Downloads

Abstract

This paper presents a mixed-mode I/II fracture criterion to investigate the crack initiation in orthotropic materials in which the crack is directed along fibers. The minimum strain energy density criterion is extended to investigate the cracked orthotropic materials. The reinforced isotropic solid model based on collinear crack propagation along fibers is proposed as an advantageous model to study the fracture behavior of composites. This model introduces fibers as reinforcements of the isotropic matrix in orthotropic materials in which their effects are qualified by defining reinforcement factors at tension and shear modes. The proposed criterion can predict the crack initiation phenomenon. Therefore, in this study a new concept of linear fracture toughness for orthotropic materials is proposed. Experimental data are used to validate the output of the proposed criterion. The coincidence of fracture limit curves and experimental data indicates the ability of the new criterion to predict crack initiation in orthotropic materials.

List of symbols

\(F_\mathrm{Y}, F_\mathrm{U}\)

Yield, ultimate force at F–D diagram

\(K_{\mathrm{Ic}}^{\mathrm{L}} ,K_{\mathrm{IIc}}^{\mathrm{L}}\)

Mode I and mode II linear fracture toughness

\(K_{\mathrm{Ic}}^{\mathrm{NL}} ,K_{\mathrm{IIc}}^{\mathrm{NL}}\)

Mode I and mode II nonlinear fracture toughness

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

Mode I and mode II fracture toughness

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

Mode I and mode II stress intensity factors

\(\rho _{\mathrm{NL}} ,\rho _{\mathrm{L}}\)

Nonlinear and linear damage factor

\(w_{\mathrm{m}} ,w_{\mathrm{f}} ,w\)

The width of matrix, fiber, RVE

\(l, \delta l\)

RVE’s Length and longitudinal displacement

t

The thickness of the RVE

\(\varepsilon _{\mathrm{m}} ,\varepsilon _{\mathrm{f}} ,\varepsilon \)

Strain of the matrix, fiber, RVE

\(\sigma _{\mathrm{m}} ,\sigma _{\mathrm{f}} ,\sigma \)

Normal stress of the matrix, fiber, RVE

\(E_{\mathrm{m}} ,E_{\mathrm{f}} ,E_{xx} ,E_{yy}\)

Elastic module of matrix, fiber, RVE along fiber, RVE across fiber

\(G_{\mathrm{m}} ,G_{\mathrm{f}} ,G_{xy} ,G_{yx}\)

Shear module of matrix, fiber, RVE in the xy plane

\(F_{\mathrm{m}} ,F_{\mathrm{f}} ,F\,\mathrm{or} F_\mathrm{c}\)

The force applied to matrix, fiber, RVE

\(A_{\mathrm{m}} ,A_{\mathrm{f}} ,A\)

The area in which the force is applied to matrix, fiber, RVE

\(\gamma _{\mathrm{m}} ,\gamma _{\mathrm{f}} ,\gamma \)

The shear angle of the matrix, fiber, RVE

\(\tau _{\mathrm{m}} ,\tau _{\mathrm{f}} ,\tau \)

Shear stress of matrix, fiber, RVE

\({\Delta }_{\mathrm{m}} ,{\Delta }_{\mathrm{f}} ,{\Delta }\)

The longitudinal displacements of matrix, fiber and RVE in pure shear loading

\(V_{\mathrm{m}} ,V_{\mathrm{f}}\)

Matrix and fiber fraction in a composite

\(E_{ij}, G_{ij} ,\nu _{ij} \)

Elastic, shear modulus and Poisson’s ratio of a composite in different directions

\({\xi }_1 ,{\xi }_2 ,{\xi }_3 \)

Reinforcement, ReSt, factors

\(\sigma _{ij}, \,\varepsilon _{ij} \)

Stress and strain functions

\(\sigma _{ij}^{\mathrm{iso}} ,\sigma _{ij}^{\mathrm{ortho}} \)

Stress state of isotropic and orthotropic materials

\(f_{ij} \left( \theta \right) ,\,g_{ij} \left( \theta \right) \)

Angular function in stress state

\(S,S_\mathrm{c} \)

Strain energy density factor, critical strain energy density factor

\(\frac{\mathrm{d}W}{\mathrm{d}V}\)

Strain energy density

\(r, \theta \)

Polar distance from the crack tip, polar angle

\(C_{ij}, \,C_{ij}^{\prime } \)

Compliance matrix for plane stress and plane strain condition

\(A_{11} ,A_{22} ,A_{12} \)

The factors in SED criterion

\(r_\mathrm{c} ,\theta _\mathrm{c} \)

Critical distance from crack tip and the path of crack growth in SED criterion

LRT

Longitudinal, radial and tangential axis in wood

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Notes

Acknowledgements

The authors would like to acknowledge the financial support of University of Tehran for this research under Grant No. 28686/01/01.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of New Sciences and TechnologiesUniversity of TehranTehranIran

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