Theory of Damping due to Thermally Assisted Unpinning of Dislocations

By algebraic methods, this paper obtains expressions for the decrement and modulus defect due to the breakaway of a dislocation from a row of equally spaced pinning points, using expressions for the activation energy given in an earlier paper. The model is that of Teutonico, Granato, and Lücke, but incorporating a pinning force of general shape. The plane, representing the complete range of values of temperature T and stress amplitude σ0, is found to consist of five regions (A to E) of qualitatively different behavior. In regions B, C, and D, breakaway (from the whole row) is activated at a single pin, as in the Granato–Lücke model; in A and E, breakaway is activated over a group of several pins. In A and B, few of the dislocations break away during a stress cycle; in C and E, nearly all do. As one moves from the "few-all" boundary into a "few" region (A or B) by decreasing T or σ0, the decrement decreases rapidly in an exponential fashion. As one moves into an "all" region (C or E), it decreases, but not rapidly. For [Formula: see text] (short loops), only two regions, A and E, exist; here, β = LcU0/Cr2, where Lc is the distance between pins, U0 is the maximum binding energy between a dislocation and a pin, C is the tension of the dislocation, and the pinning force has its maximum at a displacement of order r, the "range". Region D differs from the other regions in that there the activation energy for unpinning remains effectively constant during the cycle; the theoretical damping peak of Koiwa and Hasiguti lies in D. Region D exists only when [Formula: see text] (very long loops); here, νt is the effective attack frequency for unpinning, ω is the angular frequency of the applied stress, and the pinning force varies as y−γ−1 at large displacements y. The peak decrement, which is constant along the "few-all" boundary except near region D, drops by a factor of 2.5 when D is entered. All the boundaries between the five regions are obtained. Within each region, expressions for the decrement and modulus defect are given, which are accurate at least in limiting cases; stresses near the mechanical breakaway stress are also discussed. Generalized forms of these expressions are also given, applying to a class of damping mechanisms possessing symmetry of the breakaway type. The theory is related to experimental results of Hutchison and Rogers, and Koiwa and Hasiguti.