Brittle to Ductile Transition in Intermetallic Alloys
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BB6.3.1
Brittle to Ductile Transition in Intermetallic Alloys Jeffrey W. Kysar Department of Mechanical Engineering, Columbia University, New York, NY 10027 ABSTRACT Two distinct approaches are commonly invoked to explain the brittle to ductile transition of a material in the presence of a crack. In one approach, dislocation mobility plays the key role. In the other approach, the competition between crack tip dislocation nucleation and cleavage failure plays the key role. The two approaches are reconciled in the present study. INTRODUCTION The goal is to determine the initial energy dissipation mechanism activated at or near a crack tip. The energy dissipation mechanisms considered are: cleavage, crack tip dislocation nucleation as well as dislocation nucleation from a Frank-Read source near a crack tip. It is assumed that the brittle to ductile transition corresponds to a transition in the initial energy dissipation mechanism. The fracture criterion most commonly used for materials in which a significant amount of plastic deformation occurs is G = 2γ s + γ p , where G is the applied energy release rate available to effect fracture, γ s is the free energy of the newly created surface, and γ p is the energy dissipated through various irreversible processes in the near crack tip region. The magnitude of γ p often exceeds γ s by orders of magnitude, nevertheless it is known that γ p = γ p (γ s ) , so the energetic contribution of the newly created surfaces can not be neglected. We consider in a dimensional analysis the effect that other variables can have on γ p . It is assumed that an atomically sharp crack exists in an elastic-plastic material within which plastic deformation occurs via the creation and motion of dislocations on discrete slip planes and in discrete directions at a critical resolved shear stress. In addition to γ s , other important variables include: µ the elastic shear modulus, µ ij the so-called Schmid factor that contains information about orientations of the plastic slip systems, and σ max the maximum theoretical tensile stress that the material can support in the absence of any defect. Yield stress is also a critical parameter. However since it is a phenomenological and ill-defined variable, we instead appeal to the physics-based variables which determine yield stress: b the Burgers vector, ρ disl the density of mobile dislocations, and τ p the Peierls stress. Performing a straightforward analysis yields
γp τp σ γ 12 = F s , bρ disl , , max , µ ij . µb µ µ µb
(1)
We now explore the physical significance of each dimensionless group. The term γ s µb expresses the competition between cleavage at a crack tip and crack tip dislocation nucleation [1,2]. Its physical significance is more clearly illustrated if it is rescaled so the numerator and denominator have units of energy per length
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2γ s b < 1. µb 2 5
(2)
The numerator of Eq.(2) can be interpreted as the activation energy per unit length necessary to propagate a crack in the absence of any plastic deformation, because
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