Inverse temperature dependence of activation volume in ultrafine-grained copper processed by accumulative roll-bonding

Abstract

Tensile tests and strain-rate jump tests were carried out at several temperatures below room temperature on Cu processed by accumulative roll-bonding (ARB). The temperature dependences of the flow stress and the activation volume were determined. In contrast to conventional coarse-grained materials where the activation volume increases with increasing temperature, the ARB processed copper by 8 cycles with ultrafine-grains showed inverse temperature dependence of activation volume, i.e., decreased activation volume with increasing temperature. This inverse temperature dependence of the activation volume is discussed in terms of thermally activated dislocation bow-out from grain-boundaries.

Appendix. Derivation of Eq. 3 from Eq. 2

Eq. 2 can be rewritten as

$$ V^{\ast} /\left( {r^{2} b} \right) = \left[ {\sin^{ - 1} \left( {\eta + \delta \eta } \right) - \sin^{ - 1} \left( \eta \right)} \right] - \left[ {\left( {\eta + \delta \eta } \right)\sqrt {1 - \left( {\eta + \delta \eta } \right)^{2} } - \eta \sqrt {1 - \eta^{2} } } \right], $$

(11)

where $\eta = L_{\text{GB}} /(2r) > 0$ and $\delta \eta = w^{\ast} /(2r) > 0$. The authors also have $\eta = L_{\text{GB}} /(2r) = \sin \theta$ where θ is the critical bow-out angle. Since the activation distance w* of the pinning point is naturally much shorter than the average distance $L_{\text{GB}}$ between pinning points, the authors have $\left| {\delta \eta /\eta } \right| \ll 1$.

The function $\sin^{ - 1} \left( \eta \right)$ in the right side of Eq. 11 is given by the series written as

$$ \sin^{ - 1} \left( \eta \right) = \sum\limits_{n = 0}^{\infty } {{\frac{{\left( {2n - 1} \right)!!\eta^{2n + 1} }}{{\left( {2n} \right)!!\left( {2n + 1} \right)}}}} , $$

(12)

where $\left( {2n - 1} \right)!! = \left( {2n - 1} \right)\left( {2n - 3} \right) \ldots \, 3 \cdot 1$ and $\left( {2n} \right)!! = \left( {2n} \right)\left( {2n - 2} \right) \ldots \, 4 \cdot 2.$ When $\left| {\delta \eta /\eta } \right| \ll 1$, we have

$$ \sin^{ - 1} \left( {\eta + \delta \eta } \right) = \sum\limits_{n = 0}^{\infty } {{\frac{{\left( {2n - 1} \right)!!\left( {\eta + \delta \eta } \right)^{2n + 1} }}{{\left( {2n} \right)!!\left( {2n + 1} \right)}}}} \approx \sum\limits_{n = 0}^{\infty } {{\frac{{\left( {2n - 1} \right)!!\left[ {\eta^{2n + 1} + \left( {2n + 1} \right)\eta^{2n} \delta \eta } \right]}}{{\left( {2n} \right)!!\left( {2n + 1} \right)}}}} . $$

(13)

From Eqs. 12 and 13, we have

$$ \left[ {\sin^{ - 1} \left( {\eta + \delta \eta } \right) - \sin^{ - 1} \left( \eta \right)} \right] \approx \sum\limits_{n = 0}^{\infty } {{\frac{{\left( {2n - 1} \right)!!\eta^{2n} \delta \eta }}{{\left( {2n} \right)!!}}}} . $$

(14)

When θ is not so large, for example, θ < 1, the higher-order terms of $\eta^{2n}$, such as $\eta^{6}$, $\eta^{8}$, $\eta^{10}$,…, can be regarded much smaller than $\eta^{0} = 1$. Hence, we have

$$ \left[ {\sin^{ - 1} \left( {\eta + \delta \eta } \right) - \sin^{ - 1} \left( \eta \right)} \right] \approx \delta \eta \left( {1 + {\frac{{\eta^{2} }}{2}} + {\frac{{3\eta^{4} }}{8}}} \right). $$

(15)

When $0 < \delta \eta /\eta \ll 1$ is satisfied and the higher-order terms of $\eta^{2n}$ are neglected, we also have

$$ \left[ {\left( {\eta + \delta \eta } \right)\sqrt {1 - \left( {\eta + \delta \eta } \right)^{2} } - \eta \sqrt {1 - \eta^{2} } } \right] \approx \delta \eta \left( {1 - {\frac{{3\eta^{2} }}{2}} - {\frac{{5\eta^{4} }}{8}}} \right). $$

(16)

From Eqs. 11, 15 and 16, the authors have

$$ V^{\ast} /\left( {r^{2} b} \right) \approx \delta \eta \left( {2\eta^{2} + \eta^{4} } \right). $$

(17)

Using $\eta = L_{\text{GB}} /(2r)$, $\delta \eta = w^{\ast} /(2r)$ and $r = L_{\text{GB}} /(2\sin \theta )$, Eq. 3 can be obtained from Eq. 17.