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Plane stress asymptotic solution for steady crack growth in an elastic/perfectly plastic solid for mode I crack propagation


Previous efforts at solving an asymptotic, plane stress, steady-state crack propagation problem for a linear elastic/perfectly plastic material, under the von Mises yield condition, have failed to produce solutions that satisfy all necessary physical criteria for the mode I crack problem. These include continuous elastic and plastic stress fields that satisfy equilibrium without violating the yield condition, positive plastic work, and various jump conditions for strain rates. By using an alternative yield criterion, the parabolic von Mises, an asymptotic solution is obtained for this particular problem. This alternative yield condition was proposed by Richard von Mises as an approximation of the elliptical yield locus in the principal stress plane. It replaces the standard ellipse of the von Mises yield condition with two intersecting parabolas. The resulting yield surface has hyperbolic partial differential equations governing its entirety with the exception of two points, which are similar mathematically to the square corners of the Tresca yield surface. These corners provide the extra degree of freedom necessary to allow an asymptotic solution of this steady-state crack problem that has proven unattainable for the conventional von Mises yield condition under plane stress loading conditions.

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Unger, D.J. Plane stress asymptotic solution for steady crack growth in an elastic/perfectly plastic solid for mode I crack propagation. Z. Angew. Math. Phys. 74, 77 (2023).

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  • Parabolic von Mises yield condition
  • Steadily moving mode I crack

Mathematics Subject Classification

  • 74A45
  • 74C05