Evaluation of the incremental ERR in interface cracks with frictional contact and its application in the coupled criterion of finite fracture mechanics

A crack located in a straight and perfectly bonded interface between dissimilar isotropic linear elastic materials with a frictional contact zone adjacent to the crack tip is considered under plane strain conditions. Assuming the Coulomb friction law, the crack-tip stress singularity in such a crack is weaker than the classical square-root singularity. The main difficulty in predicting propagation of such an interface crack is that the Energy Release Rate (ERR) is zero, which is a direct consequence of this weak stress singularity at the crack tip. Therefore, the Griffith fracture criterion, which assumes infinitesimal crack advances, cannot be applied in this case. To overcome this problem a new approach to predict the propagation of an interface crack with a frictional contact zone at the crack tip, based on the Coupled stress and energy Criterion (CC) of Finite Fracture Mechanics (FFM), is proposed and analyzed. In contrast to previous approaches, the critical finite crack advance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:msub> </mml:mrow> </mml:math> is determined by the CC as a structural parameter given by the overall problem configuration. Two methods for calculating the incremental ERR <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi mathvariant="normal">II</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are considered which differ in the treatment of the frictional energy dissipated along the crack advance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> . Closed-form expressions for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi mathvariant="normal">II</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are derived for sufficiently large interface cracks when the most singular term of the asymptotic expansion of the elastic solution at the crack tip is dominant along the path of crack advance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> before the crack propagation occurs. In this case, closed-form expressions for the critical crack advance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:msub> </mml:mrow> </mml:math> and the critical stress intensity factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi mathvariant="normal">IIc</mml:mi> </mml:msub> </mml:math> are derived.