Analysis on the Dependence of the Fracture Locus on the Pressure and the Lode Angle

  • Nouira Meriem
  • Oliveira Marta
  • Khalfallah AliEmail author
  • Alves José
  • Menezes Luís
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


Classical fracture models assume that the stress triaxiality is the key parameter controlling the magnitude of the fracture strain. However, recent works shown the influence of other parameters that characterize the stress state on the prediction of fracture strains. In this work, two uncoupled fracture models, Mae and Wierzbicki [8] and Xue and Wierzbicki [9], were analysed using finite element models. These models define a ductile fracture locus formulated in the 3D space of the stress triaxiality, Lode angle parameter and the equivalent fracture strain. The material selected was a cast A356 aluminium alloy for which the model parameters were previously defined. Two groups of tests are analysed in order to provide additional information on the material ductility. The first corresponds to plane strain tests carried out on flat plates with different grooves. The second corresponds to uniaxial tension tests applied on smooth and notched round bars, which were designed with different notch radii. These specimens allow covering a wide range of stress triaxiality. The present work extracts the evolution of the equivalent plastic strain at fracture, the stress triaxiality and the Lode angle parameter in order to evaluate the possibility of using either smooth and notched bars tests or smooth and notched bars tests and grooved plates to evaluate the 3D locus for high values of stress triaxiality. In this context, a new function is proposed to describe the equivalent plastic strain at fracture based on the stress triaxiality and the Lode angle parameter.


Uncoupled fracture models 3D fracture locus Equivalent fracture strain Stress triaxiality Lode angle 


\( {E} \)

Young’s modulus

\( {k} \)

Exponent in curvilinear Lode angle dependence function

\( {m} \)

Damage exponent

\( {n} \)

Strain hardening exponent

\( {p} \)

Mean pressure

\( {p}_{{{lim}}} \)

Limiting pressure below which no damage occurs

\( {q} \)

Exponent in pressure dependence function

\( {\eta}= \sigma_{m} /\sigma_{eq} \)

Stress Triaxiality

\( \bar{{\varepsilon }}_{{f}} \)

Equivalent fracture strain

\( \bar{{\varepsilon }}_{{{f}0}} \)

Reference equivalent fracture strain

\( ({\bar{{\upvarepsilon}}}_{{{\mathbf{f}},{\mathbf{t}}}} ) \)

Effective fracture strain under uniaxial tension

\( \left( {{\bar{{\upvarepsilon}}}_{{{\mathbf{f}},{\mathbf{s}}}} } \right) \)

Effective fracture strain under pure shear

\( {\gamma} \)

Ratio of fracture strains

\( {\xi}, {\theta}_{{l}} \)

Third invariant of the deviatoric stress tensor, Lode angle

\( {\mu}_{{p}} \)

Pressure dependence function

\( {\mu}_{{\theta}} \)

Lode angle dependence function

\( \vartheta \)

Poisson’s ratio

\( {\sigma}_{1,2,3} \)

Principal components of the Cauchy stress tensor

\( {\sigma} \)

Cauchy Stress tensor

\( {\sigma}_{{m}} \)

Mean stress

\( {\sigma}_{{{eq}}} ,\bar{{\sigma }} \)

Equivalent stress

\( {J}_{3} = {\text{s}}_{1} {\text{s}}_{2} {\text{s}}_{3} \)

Is the third stress invariant

\( {D} \)

Damage accumulation

\( {y}_{0} , {k}\,{\text{and}}\, {n} \)

Swift law hardening parameters.


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Nouira Meriem
    • 1
    • 2
  • Oliveira Marta
    • 3
  • Khalfallah Ali
    • 1
    • 4
    Email author
  • Alves José
    • 5
  • Menezes Luís
    • 3
  1. 1.Laboratoire de Génie MécaniqueEcole Nationale d’Ingénieurs de Monastir, University of MonastirMonastirTunisia
  2. 2.Ecole Nationale d’Ingénieurs de Sousse, University of SousseSousseTunisia
  3. 3.Department of Mechanical EngineeringCEMMPRE, University of CoimbraCoimbraPortugal
  4. 4.Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, University of SousseSousseTunisia
  5. 5.Department of Mechanical EngineeringCMEMS, University of MinhoGuimarãesPortugal

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