Determination of the internal stress and dislocation velocity stress exponent with indentation stress relaxation test
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dentation stress relaxation tests were carried out on high-purity polycrystalline copper specimens at room temperature with a flat cylindrical indenter. The experimental results showed that the resulting load-time relaxation curves can be described by a power law, which coupled an internal stress and an integral constant between the effective stress and relaxation time. Then the internal stress, integral constant, and dislocation velocity stress exponent can be extracted from load relaxation curves. The derived values from this way were consistent with the results of conventional uniaxial compression stress relaxation tests. These agreements are not only useful to understand deformation (dislocation) mechanisms under the indenter, but also exhibit an attractive potential of measuring nano/micromechanical properties of materials by indentation test.
I. INTRODUCTION
The relationship between the applied stress and dislocation velocity is a central concept of the microdynamical theory of the crystal plasticity and has received considerable attention in the past few decades.1–4 In general, the applied stress needed to induce crystal deformation can be broken down into an internal stress i and effective stress (also called thermal stress) *, that is: = i + *
(1)
.
Here, we define i as the minimum stress (corresponding to zero strain rate) of sustaining a crystal plastic deformation. The effective stress *, acting on dislocations during their thermally activated motion, is dependent on the temperature and strain rate. Due to the central status of internal stress i in revealing the deformation mechanism and computing other deformation parameters, several methods have been explored to determine the internal stress experimentally,5–9 and the stress relaxation test is one of the most attractive approaches among them. It is known that the usual relation between the stress and relaxation time t is a power law relation6: − i = K(t + a)−n
,
(2)
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2008.0299 2486
http://journals.cambridge.org
J. Mater. Res., Vol. 23, No. 9, Sep 2008 Downloaded: 02 Apr 2015
where K is a constant that relates to the stiffness of the testing system and the density of mobile dislocation, a is an integrating constant, and n ⳱ 1/(m − 1) (m is a stress exponent of dislocation velocity). Thus the stress rate d/dt is: −
d = nK共t + a兲− n −1 dt
.
(3)
Substituting Eq. (2) to Eq. (3), we have: −
d = nK −1 Ⲑ n 共 − i 兲m dt
.
(4)
From the aforementioned analysis, the integrating constant a, internal stress i, and dislocation velocity stress exponent m can be obtained from stress relaxation tests. For the materials with limitations of structure and volume such as thin-film systems, their mechanical properties display a significant difference from bulks counterpart and cannot be measured by the conventional compression or tensile stress relaxation tests. As an alternative approach, the indentation technique, which is evolved from hard
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