Effect of crack tip shape on near-tip deformation and fields in plastically compressible solids

  • Md Intaf Alam
  • Debashis KhanEmail author
  • Yash Mittal
  • Sandeep Kumar
Technical Paper


In the present study, the crack tip shape effect on near-tip deformation and fields is numerically investigated for a mode I crack under plane strain and small-scale yielding conditions. We explore here the quasi-static deformations of solids characterized by finite strain elastic–viscoplastic material model with bilinear hardening and hardening–softening–hardening hardness functions. For comparative analyses, both plastically incompressible and plastically compressible solids have been considered. It has been observed that the crack tip shape can have great consequence on the near-tip deformation and plastic fields. As the crack tip radius is increased, the plastic strain and stresses advance more to the tip of a crack as compared to the crack surface. It has also been revealed that the combination of crack tip curvature radius, material softening and plastic compressibility provides some useful and fundamental information for the near-tip deformation and plastic fields.


Mode I crack Crack tip shape Finite deformation Compressible solid Plasticity 

List of symbols


Semi-major axis of the elliptical crack tip


Semi-minor axis of the elliptical crack tip


Crack tip extension


Rate of deformation tensor


Elastic part of rate of deformation tensor


Plastic part of rate of deformation tensor


Young’s modulus


Deformation gradient


Hardness function

h1, h2, h3

Slopes of the tri-linear hardness function


Identity tensor




Applied J-integral


Mode I stress intensity factor


Tensor of elastic moduli


Hardening exponent


Deviatoric part of Kirchhoff stress tensor


Outer radius of the semicircular geometry




Vertically aligned carbon nanotubes


Convected coordinates

Greek symbols


Plastic compressibility


Cauchy stress tensor


Kirchhoff stress


Poisson’s ratio

\({\hat{\mathbf{\tau }}}\)

Jaumann rate of Kirchhoff stress


Plastic strain


Plastic strain rate

\(\dot{\varepsilon }_{0}\)

Reference strain rate


Effective stress


Reference stress

σxx, σyy

Normal stresses in x and y directions, respectively


Shear stress


Hydrostatic stress


Equivalent stress



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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology (BHU) VaranasiVaranasiIndia

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