A thermodynamic analysis of the empirical power relationships for creep rate and rupture time
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I.
εz 5 C2s n2, n2 . 0
INTRODUCTION
TWO-PARAMETRIC
power relationships are often used in the material science in order to approximate the experimental data of time-dependent characteristics of strength and fracture. The form of the empirical power relationship is usually given as follows y 5 Ci x ni, i51,2,3...
[1]
where the y and x are the values of the dependent variable and the argument, respectively, and Ci and ni are usually called materials constants. Several examples where Eq. [1] is used are listed as follows. A. Isothermal Steady-State Creep and Creep Rupture Time Equation [1] is used[1–9,11–14,18] for isothermal steady-state creep in the form
εz 5 C1s n1, ...n1 . 0
[2]
where εz is the steady-state (minimum) creep rate at a given value of stress, s 5 const, and C1 and n1 are empirical parameters, or in the form[1,3–5,8,10,12,13,15,16] tt 5 C'1s n1, ...n'1 , 0
[3]
where tt is the time to rupture and s is the rupture stress. There is also established the relation between εz and tt (the Monkman–Grant rule): n"1 εz 5 C"t 1 t , n"1 ' 21
[4]
where C "1, and n"1 , are the empirical constants. B. Isothermal Tensile Testing at Constant Strain Rate Equation [1] can be used[23,24,25] for the moderate and low strain rates in the form
where s designates the stress corresponding to a given deformation (e.g., yield stress). The majority of cited references use a linearization of Eqs. [1] through [5] in order to estimate the empirical parameters Ci and ni. In this case, Eq. [1] is transformed into a straight line of the form log y 5 ni log x 1 log Ci
METALLURGICAL AND MATERIALS TRANSACTIONS A
[6]
where ni and log Ci are the slope and the log y value at log x 5 0, respectively. Stocker and Ashby[19] analyzed the steady-state creep data for a number of pure metals, alloys, carbides, and ionic materials. They established a clear correlation between the dimensionless material parameters n and A from the semiempirical Dorn[20] equation:
~!
DGb s εz 5 A kT G
n
[7]
in which D is the effective diffusivity, G is the appropriate shear modulus, b is the Burger’s vector, k is the Boltzman’s constant, and T is absolute temperature. This correlation has a form lgA 5 210.5 1 3.4n
[8]
It is almost independent of the material considered and will be discussed later. The main goal of our work is to establish the general relationships between the aforementioned sets of experimental data and to explain why they have similar powerlaw approximations (Eqs. [1] through [5]). In this article, we discuss three sets of data: (1) steady-state creep, Eq. [2]; (2) creep rupture time, Eq. [3]; and (3) tensile tests at constant strain rate, Eq. [5]. II.
A.J. KRASOWSKY, Professor and Department Head, is with the Institute for Problems of Strength, National Academy of Science of Ukraine, Kiev 252014, Ukraine. L. TOTH, Professor, is with the Department of Mechanical Engineering, University of Miskolc, Miskolc, H-3515 Hungary. Manuscript submitted September 8, 1995.
[5]
RESULTS
Not only is the power two-parametrical character of Eq.
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