- Email: [email protected]
Residual strain and the ‘fracture stress-temperature’ relation in beryllium oxide
JOURNAL
OF NUCLEAR MATERIALS
RESIDUAL
STRAIN
14 (1964)203-204@
AND
THE
RELATION T. W. BAKER, Metallurgy
NORTH-HOLLAND
‘FRACTURE
PUBLISHING
STRESS-TEMPERATURE’
IN BERYLLIUM
OXIDE
P. J. BALDOCK
and F. J P. CLARKE
Division,
Havwell,
AERE,
It is shown that the magnitude of residual strain due to anisotropic lattice expansion is sufficient to account
Beds.,
UK
for an experimental form of ‘fracture stress-temperature’ relation that has been reported in the literature.
--_
1. Introduction
In a previous paper l) an expression was derived for the influence of residual strain on the ‘fracture strength-grain size’ relationship in brittle solids. The basic difficulty in testing the model lies in obtaining samples of different grain sizes in which other parameters such as magnitude and type of porosity are fixed. However, at constant grain size the model also predicts a form of ‘fracture stress-temperature’ relationship in so far as it depends on residual strain. Experimental results on this relationship are available “) and to compare them with the predictions of the model, values of the lattice parameters as a function of temperature are needed. These have been obtained experimentally and a preliminary curve sufficient for the immediate purpose is reported in this note; the theoretical and experimental forms of ‘fracture stress-temperature’ relationship are compared. A detailed report of the X-ray studies is given elsewhere 3). 2. Comparison
CO. AMSTERDAM
EXPERIMENTAL THEORETICAL
1I
-0
t
4.35
3.60 t
100
500 TEMP.
1000 *c
Fig. 1. (top) theoretical and experimental *) forms of ‘u&r against temperature’. Dotted curves are the theoretical values for three different grain sizes. Grain sizes were not quoted in reference 2. (bottom) ‘c’ and ‘a’ lattice parameters as a function of temperature. Values plotted are:
of Theory and Experiment
The basic eq. (1) is
Temp. “C)
‘a’ (A)
‘c’ (A)
21 209 626
2.6996 2.7039 2.7117 2.7173 2.7268
4.3770 4.3837 4.3966 4.4038 4.4169
760
980 E =
residual strain subscripts 1 and 2 refer to two temperatures III.
RADIATION
EFFECTS
204 E = 21 = c0 = v = y =
T. W. BAKER, P. J. BALDOCK AND F. J. P. CLARKE
Young’s modulus, taken as 45 x lo6 psi Average grain diameter Average pore diameter Poisson’s ratio, taken as 0.34 surface energy in the absence of residual strain, taken as 760 ergs. cm-a l),
The variation of the ‘c’ and ‘LZ’lattice parameters as a function of temperature is shown at the bottom of fig. 1. These measurements were made on the solid material but it was checked that at room temperature the values were sufficiently close to those obtained on powdered material to make no difference to the calculation. In obtaining E from these curves it has been assumed that all differential thermal contraction below 1000” C is accommodated in a purely elastic manner. The top curves on fig. 1 show the ratio a,/~,, where subscript 2 refers to the temperature on the abscissa and subscript 1 refers to room temperature. Values predicted by the equation above, for three different grain sizes, are com-
pared with some experimental values from “). The authors in reference 2 do not quote grain size but the range 50-100,~ is reasonable. 3* Discussion The predicted and experimental values agree very well up to 700-800” C. In this range the experimental curve “) begins to drop off. The reason for this is not known; it may be associated with loss of grain boundary cohesion. The good agreement between theory and experiment is no proof that residual strain is the factor determining the shape of the fracture stress-temperature curve. All that can be said is that the magnitude of the residual strain is, in the absence of other mechanisms, sufficient to account for it.
References 1) F. J. P. Clarke,
Acta. Met. 12 (1964) 139 2) M. L. Steksel, R. M. Hale and C. E. Waller, Mechanical Properties of Engineering Ceramics (Interscience, 1961) 226 8) T. W. Baker and P. J. Baldock (in course of publication)
OF NUCLEAR MATERIALS
RESIDUAL
STRAIN
14 (1964)203-204@
AND
THE
RELATION T. W. BAKER, Metallurgy
NORTH-HOLLAND
‘FRACTURE
PUBLISHING
STRESS-TEMPERATURE’
IN BERYLLIUM
OXIDE
P. J. BALDOCK
and F. J P. CLARKE
Division,
Havwell,
AERE,
It is shown that the magnitude of residual strain due to anisotropic lattice expansion is sufficient to account
Beds.,
UK
for an experimental form of ‘fracture stress-temperature’ relation that has been reported in the literature.
--_
1. Introduction
In a previous paper l) an expression was derived for the influence of residual strain on the ‘fracture strength-grain size’ relationship in brittle solids. The basic difficulty in testing the model lies in obtaining samples of different grain sizes in which other parameters such as magnitude and type of porosity are fixed. However, at constant grain size the model also predicts a form of ‘fracture stress-temperature’ relationship in so far as it depends on residual strain. Experimental results on this relationship are available “) and to compare them with the predictions of the model, values of the lattice parameters as a function of temperature are needed. These have been obtained experimentally and a preliminary curve sufficient for the immediate purpose is reported in this note; the theoretical and experimental forms of ‘fracture stress-temperature’ relationship are compared. A detailed report of the X-ray studies is given elsewhere 3). 2. Comparison
CO. AMSTERDAM
EXPERIMENTAL THEORETICAL
1I
-0
t
4.35
3.60 t
100
500 TEMP.
1000 *c
Fig. 1. (top) theoretical and experimental *) forms of ‘u&r against temperature’. Dotted curves are the theoretical values for three different grain sizes. Grain sizes were not quoted in reference 2. (bottom) ‘c’ and ‘a’ lattice parameters as a function of temperature. Values plotted are:
of Theory and Experiment
The basic eq. (1) is
Temp. “C)
‘a’ (A)
‘c’ (A)
21 209 626
2.6996 2.7039 2.7117 2.7173 2.7268
4.3770 4.3837 4.3966 4.4038 4.4169
760
980 E =
residual strain subscripts 1 and 2 refer to two temperatures III.
RADIATION
EFFECTS
204 E = 21 = c0 = v = y =
T. W. BAKER, P. J. BALDOCK AND F. J. P. CLARKE
Young’s modulus, taken as 45 x lo6 psi Average grain diameter Average pore diameter Poisson’s ratio, taken as 0.34 surface energy in the absence of residual strain, taken as 760 ergs. cm-a l),
The variation of the ‘c’ and ‘LZ’lattice parameters as a function of temperature is shown at the bottom of fig. 1. These measurements were made on the solid material but it was checked that at room temperature the values were sufficiently close to those obtained on powdered material to make no difference to the calculation. In obtaining E from these curves it has been assumed that all differential thermal contraction below 1000” C is accommodated in a purely elastic manner. The top curves on fig. 1 show the ratio a,/~,, where subscript 2 refers to the temperature on the abscissa and subscript 1 refers to room temperature. Values predicted by the equation above, for three different grain sizes, are com-
pared with some experimental values from “). The authors in reference 2 do not quote grain size but the range 50-100,~ is reasonable. 3* Discussion The predicted and experimental values agree very well up to 700-800” C. In this range the experimental curve “) begins to drop off. The reason for this is not known; it may be associated with loss of grain boundary cohesion. The good agreement between theory and experiment is no proof that residual strain is the factor determining the shape of the fracture stress-temperature curve. All that can be said is that the magnitude of the residual strain is, in the absence of other mechanisms, sufficient to account for it.
References 1) F. J. P. Clarke,
Acta. Met. 12 (1964) 139 2) M. L. Steksel, R. M. Hale and C. E. Waller, Mechanical Properties of Engineering Ceramics (Interscience, 1961) 226 8) T. W. Baker and P. J. Baldock (in course of publication)