.. _section-A9.5:

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]

.. math::
   :label: fuel_E

    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}

.. math::
   :label: clad_E

    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}

**Poisson's Ratio** (-)

.. math::
   :label: fuel_p

    \nu = \begin{cases}
       0.3  & \text{phase = } \alpha + \delta \\
       0.3  & \text{phase = } \beta + \gamma \text{ or } \gamma \\
    \end{cases}

.. math::
   :label: clad_Ep

    \nu = \begin{cases}
        0.3  & \text{Clad = HT9 } \\
        0.3  & \text{Clad = D9 } \\
    \end{cases}

**Thermal Expansion Coefficienti** (:math:`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 :math:`T_0=293.15` K, is

.. math::
   :label: clad_alpha

    \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}

**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 :numref:`table-A9.5-1`.

.. _table-A9.5-1:

.. list-table:: Selected reference point for pore sintering yield stress
    :header-rows: 1
    :align: center
    :widths: auto

    * - 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 (:math:`\alpha_{p}`), then
solves the following equation to compute the pore yield
strength(:math:`\sigma_{f})`:

.. math::
   :label: eq158

    \epsilon_{f}(\sigma_{f})\alpha_{pf} = \epsilon_{ref}(\sigma_{ref})\alpha_{pref}

where :math:`\epsilon_{f}` is the fuel equivalent creep rate (1/s)
given the current temperature and hydrostatic stress,
:math:`\alpha_{pf}` is the current fuel porosity compressibility
factor (See :eq:`eq87`), :math:`\epsilon_{ref}` is the equivalent creep
rate (1/s) computed using the parameters in Table 1,
:math:`\alpha_{pref}` is the pore compressibility factor given in
Table 1, and\ :math:`\ \sigma_{f}` and :math:`\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:

.. math::
   :label: eq159

   \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 :math:`\varepsilon_{irHT9}` is the HT9 irradiation creep strain
rate (1/s), b0 =\ :math:`1.83 \times 10^{- 4}`, a =
:math:`2.59 \times 10^{14}`, q = :math:`7.3 \times 10^{4}`, T is the
temperature (K), R is the universal gas constant (cal/mol/K),
:math:`\phi` is the neutron flux (#/cm\ :sup:`2`-s/10\ :sup:`22`),
:math:`\sigma_{eq}` is the equivalent stress rate (Pa), and
:math:`D_{c}` is dose conversion (dpa/n/cm\ :sup:`2`/10\ :sup:`22`) [9-30].

D9 Clad:

.. math::
   :label: eq160

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

.. math::
   :label: eq161

    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}

Where :math:`\varepsilon_{irD9}` is the D9 irradiation creep strain
rate, T is the temperature (K), :math:`\phi` is the neutron flux
(#/cm\ :sup:`2`-s/10\ :sup:`22`), :math:`\sigma_{eq}` is the equivalent
stress rate (Pa), and D\ :sub:`c` is dose conversion
(dpa/n/cm\ :sup:`2`/10\ :sup:`22`) [:numref:`section-9.8.2.4`].

**Clad Thermal Creep**

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

HT9 Clad [9‑30]:

.. math::
   :label: eq162

   \varepsilon_{ThHT9} = \ (\varepsilon_{pHT9} + \varepsilon_{sHT9} + \varepsilon_{tHT9}) \times 0.01

.. math::
   :label: eq163

   \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}

.. math::
   :label: eq164

   \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}

.. math::
   :label: eq165

   \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}

Where T is the temperature (K), R is gas constant (cal/mol/K),
:math:`\phi` is the neutron flux (#/cm\ :sup:`2`-s/10\ :sup:`22`),
:math:`\sigma_{eq}` is the equivalent stress rate (Pa), and t is time
(s). :numref:`table-A9.5-2` includes the values of the parameters corresponding to
:eq:`eq163`, :eq:`eq164`, :eq:`eq165`.

.. _table-A9.5-2:

.. list-table:: HT9 thermal creep parameters
    :header-rows: 0
    :align: center
    :widths: auto

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

D9 Clad:

.. math::
   :label: eq166

   \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 :math:`\varepsilon_{ThD9}` is the D9 thermal creep rate (1/s), T
is the temperature (K), :math:`\sigma_{eq}` is the equivalent stress
rate (Pa), and :math:`\varepsilon_{OS}` is 38633 (1/s), m is 5.35,
:math:`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:

.. math::
   :label: eq167

    \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}

where :math:`\varepsilon_{swHT9}` is linear incremental HT9 void
swelling strain, :math:`T` is the temperature at the clad midwall (K),
:math:`\phi` is the neutron flux (#/cm\ :sup:`2`-s/10\ :sup:`22`),
:math:`\psi` is the neutron fluence (#/cm\ :sup:`2`/10\ :sup:`22`), and
D\ :sub:`c` is dose conversion (dpa/n/cm\ :sup:`2`/10\ :sup:`22`).

D9 Clad:

[9-31]

.. math::
   :label: eq168


   \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}