9.8.5. Appendix 9.5: Mechanical Properties

The elastic and thermal expansion properties for metal fuel [9-22] [9-28], HT9 and D9 cladding [9-29], with temperature defined in Kelvin, are

Elastic Modulus (Pa)

[9-12]

(9.8-20) $$\begin{split} E = \begin{cases} \left( 56 + 0.1158 \left( 865 - T \right) \right) 10^{9} & \text{phase = } \alpha + \delta \\ \left( 56 + 0.2158 \left( 865 - T \right) \right) 10^{9} & \text{phase = } \beta + \gamma \text{ or } \gamma \\ \end{cases}\end{split}$$

(9.8-21) $$\begin{split} E = \begin{cases} \left( 2.137E+05 - 102.74 \left(T - 273.15\right) \right) 10^{6} & \text{Clad = HT9 } \\ \left( 2.137E+05 - 102.74 \left(T - 273.15\right) \right) 10^{6} & \text{Clad = D9 } \\ \end{cases}\end{split}$$

Poisson’s Ratio (-)

(9.8-22) $$\begin{split} \nu = \begin{cases} 0.3 & \text{phase = } \alpha + \delta \\ 0.3 & \text{phase = } \beta + \gamma \text{ or } \gamma \\ \end{cases}\end{split}$$

(9.8-23) $$\begin{split} \nu = \begin{cases} 0.3 & \text{Clad = HT9 } \\ 0.3 & \text{Clad = D9 } \\ \end{cases}\end{split}$$

Thermal Expansion Coefficienti ($m/\Delta m-K$)

The composition-dependent thermal expansion coefficient for metallic fuel is determined by interpolating between alloy/component data in the Metallic Fuels Handbook [9-22].

The cladding thermal expansion coefficient, with temperature defined in Kelvin, and $T_0=293.15$ K, is

(9.8-24) $$\begin{split} \alpha = \begin{cases} \left(-0.2191 + 5.678E-4\left(T-T_0\right) + 8.111E-7\left(T^2-T_0^2\right) \right. \\ \left. - 2.576E-10\left(T^3-T_0^3\right)\right)10^{-2} & \text{Clad = HT9 } \\ \left(-0.4274 + 1.282E-3\left(T-T_0\right) + 7.362E-7\left(T^2-T_0^2\right) \right. \\ \left. - 2.069E-10\left(T^3-T_0^3\right)\right)10^{-2} & \text{Clad = D9 } \\ \end{cases}\end{split}$$

Fuel Pore Sintering Yield Stress

Experimental data, although limited, shows evidence of pore sintering due to the softness of metallic fuel at elevated temperatures [9‑24]. Prior to eutectic formation, any fuel expansion, caused by thermal expansion or fission product swelling, is balanced by pore sintering and fuel clad mechanical interaction. This balance is also evident in high-level experiments such as TREAT M-Series [9‑25] and Whole Pin Furnace tests [9‑26].

In order to account for the impact of pore sintering at elevated temperatures, a pore yield strength model, which is a function of creep rate and pore compressibility factor, is developed based on a reference data point in Ref. [9‑24] and expert judgement. The selected reference point for pore yield strength is given in Table 9.8.13.

Table 9.8.13 Selected reference point for pore sintering yield stress

Reference Parameters	Reference Values
Temperature	973.15 K
Hydrostatic Stress	2.5 MPa
Pore Compressibility factor	C/6
C - Fitting Factor	10

Given temperature, hydrostatic stress, and fuel porosity, the model computes the pore compressibility factor ($\alpha_{p}$), then solves the following equation to compute the pore yield strength($\sigma_{f})$:

(9.8-25) $$\epsilon_{f}(\sigma_{f})\alpha_{pf} = \epsilon_{ref}(\sigma_{ref})\alpha_{pref}$$

where $\epsilon_{f}$ is the fuel equivalent creep rate (1/s) given the current temperature and hydrostatic stress, $\alpha_{pf}$ is the current fuel porosity compressibility factor (See Eq. (9.2-87)), $\epsilon_{ref}$ is the equivalent creep rate (1/s) computed using the parameters in Table 1, $\alpha_{pref}$ is the pore compressibility factor given in Table 1, and$\ \sigma_{f}$ and $\sigma_{ref}$ are the fuel pore yield strength (MPa) and reference stress (MPa), respectively. The assumed upper limits for pore yield strength are for porous fuel with more than 10% fuel porosity and low porosity fuel with less than 10% fuel porosity are 50 MPa and 80 MPa, respectively.

Clad Irradiation Creep

Irradiation creep strain of HT9 and D9 cladding is modeled using the following equations.

HT9 Clad:

(9.8-26) $$\varepsilon_{irHT9} = b0 + a \times exp\left( - \frac{q}{RT} \right) \times \phi \times \left( \sigma_{eq} \times 10^{- 6} \right)^{1.3} \times \frac{0.01 \times D_{c}}{5}$$

where $\varepsilon_{irHT9}$ is the HT9 irradiation creep strain rate (1/s), b0 =$1.83 \times 10^{- 4}$, a = $2.59 \times 10^{14}$, q = $7.3 \times 10^{4}$, T is the temperature (K), R is the universal gas constant (cal/mol/K), $\phi$ is the neutron flux (#/cm²-s/10²²), $\sigma_{eq}$ is the equivalent stress rate (Pa), and $D_{c}$ is dose conversion (dpa/n/cm²/10²²) [9-30].

D9 Clad:

(9.8-27) $$\varepsilon_{irD9} = A_{mod} \times \phi \times \left( \sigma_{eq} \times \ 10^{- 6} \right) \times 0.01 \times D_{c}$$

(9.8-28) $$\begin{split} A_{mod} = \begin{cases} 2 \times 10^{- 6} & T \leq 723.15\ K \\ 2 \times 10^{- 6} + \frac{\left( 3 \times 10^{- 7} - 2 \times 10^{- 6} \right)}{773.15 - 723. 15}\ (T - 723.15) & 723.15 < T \leq 773.15 \\ 3 \times 10^{- 7} & T > \ 773.15 \\ \end{cases}\end{split}$$

Where $\varepsilon_{irD9}$ is the D9 irradiation creep strain rate, T is the temperature (K), $\phi$ is the neutron flux (#/cm²-s/10²²), $\sigma_{eq}$ is the equivalent stress rate (Pa), and D_c is dose conversion (dpa/n/cm²/10²²) [Section 9.8.8.2.4].

Clad Thermal Creep

Thermal creep strain of HT9 and D9 cladding is modeled using the following equations.

HT9 Clad [9‑30]:

(9.8-29) $$\varepsilon_{ThHT9} = \ (\varepsilon_{pHT9} + \varepsilon_{sHT9} + \varepsilon_{tHT9}) \times 0.01$$

(9.8-30) $$\begin{split}\varepsilon_{pHT9} = \ \left\lbrack C_{1} \times \ EXP( - \frac{Q_{1}}{RT})\ \times \ (\sigma_{eq} \times \ 10^{- 6})\ + \ C_{2}\ \times \ EXP( - \frac{Q_{2}}{RT})\ \times \ {(\sigma_{eq} \times 10 ^{- 6})}^{4}\ \right. \\ \left. + C_{3}\ \times \ \ EXP( - \frac{Q_{1}}{RT}) \ \times \ {(\sigma_{eq} \times \ 10^{- 6})}^{0.5}\ \right \rbrack \times \ EXP( - C_{4}\ \times \ t)\ \times \ C_{4}\end{split}$$

(9.8-31) $$\varepsilon_{sHT9}\ = C_{5}\ \times \ EXP( - \frac{q4}{RT})\ \times \ {(\sigma_{eq} \times \ 10^{- 6})}^{2}\ + \ C_{6}\ \ EXP( - \frac{q5}{RT})\ \times \ {(\sigma_{eq} \times \ 10^{- 6})}^{5}$$

(9.8-32) $$\begin{split}\varepsilon_{tHT9}\ = \begin{cases} C_{HT9Creep} \times 4 \times \ C_{7}\ \times \ EXP\left( - \frac{Q_{6}}{RT} \right) \times \ \left( \sigma_{eq} \times \ 10^{- 6} \right)^{10} \times \ t^{3} & T \leq 1200\ K \\ 0 & T > 1200\ K \end{cases}\end{split}$$

Where T is the temperature (K), R is gas constant (cal/mol/K), $\phi$ is the neutron flux (#/cm²-s/10²²), $\sigma_{eq}$ is the equivalent stress rate (Pa), and t is time (s). Table 9.8.14 includes the values of the parameters corresponding to Eq. (9.8-30), Eq. (9.8-31), Eq. (9.8-32).

Table 9.8.14 HT9 thermal creep parameters

$C_{1}$	13.4
$C_{2}$	8.43e-03
$C_{3}$	4.08e+18
$C_{4}$	1.6e-06
$C_{5}$	1.17e+9
$C_{6}$	8.33e+9
$C_{7}$	9.53e+21
$Q_{1}$	15027
$Q_{2}$	26451
$Q_{3}$	89167
$Q_{4}$	83142
$Q_{5}$	108276
$Q_{6}$	282700
$C_{HT9Creep}$	0.15

D9 Clad:

(9.8-33) $$\varepsilon_{ThD9} = \varepsilon_{OS}\ \times \ \left( \frac{\sigma_{eq}}{(20000\ - \ 9.12\ \times \ T)\ \times \ (92000\ - \ 40.2\ \times \ T)} \right)^{m} \times {exp}\left( - \frac{Q_{r}}{T} \right)$$

Where $\varepsilon_{ThD9}$ is the D9 thermal creep rate (1/s), T is the temperature (K), $\sigma_{eq}$ is the equivalent stress rate (Pa), and $\varepsilon_{OS}$ is 38633 (1/s), m is 5.35, $Q_{r}$ is 1.062e+14 K.

Irradiation Induced Void Swelling

Irradiation induced void swelling strain of HT9 and D9 are modeled using the following equations.

HT9 Clad:

If the cladding dose is less than 100 dpa, HT9 void swelling rate is set to zero [9-11]. Above 100 dpa, the following temperature dependent linear void swelling rate is adopted:

(9.8-34) $$\begin{split} \varepsilon_{swHT9} = \begin{cases} \left( 0.0000833 + \frac{0.0001 - 0.0000833}{50.0}*(T - 623.0) \right) \times \frac{\phi \times D_{c}}{3} & T\ \leq 673\ K \\ \left( 0.0001 + \frac{0.0000833 - 0.0001}{50.0}*(T\ - \ 673.0) \right) \times \frac{\phi \times D_{c}}{3} & 673\ K < T\ \leq 723\ K \\ \left( 0.0000833 + \frac{0.00005\ - \ 0.0000833}{50.0}*(T - 723.0) \right) \times \frac{\phi \times D_{c}}{3} & 723\ K < \ T\ \leq 773\ K \\ \left( 0.000075 + \frac{0.000025 - 0.000075}{50.0}*(T - 773.0) \right) \times \frac{\phi \times D_{c}}{3} & 773\ K < T\ \leq 823\ K \\ \left( 0.00005 + \frac{0.0000125 - 0.00005}{50.0}*(T - 823.0) \right) \times \frac{\phi \times D_{c}}{3} & 823\ K < T\ \leq 873\ K \\ 0 & T > 873\ K \end{cases}\end{split}$$

where $\varepsilon_{swHT9}$ is linear incremental HT9 void swelling strain, $T$ is the temperature at the clad midwall (K), $\phi$ is the neutron flux (#/cm²-s/10²²), $\psi$ is the neutron fluence (#/cm²/10²²), and D_c is dose conversion (dpa/n/cm²/10²²).

D9 Clad:

[9-31]

(9.8-35) $$\begin{split}\varepsilon_{swD9} = \begin{cases} 0.2 \times \frac{0.01}{3.0} \times \phi \times D_{c} & T < 723\ K\ and\ dpa\ \geq 55 \\ 0.5 \times \frac{0.01}{3.0} \times \ \phi \times D_{c} & T\ \geq 723\ K\ and\ dpa\ \geq 60 \\ 0 & \text{else} \end{cases}\end{split}$$