.. _section-8.7:

Material Properties
-------------------

All material property data are cast as functions or subroutines to allow
for modularization and the ease of making changes. This also allows for
the incorporation of different materials data in a straightforward
manner. In a number of the correlations used, the units are inconsistent
with the SI unit system adopted by SAS4A. The routines that use these
correlations carry out appropriate units conversions internally.

.. _section-8.7.1:

Fuel Density
~~~~~~~~~~~~

The solid fuel density is assumed to have the functional form

.. math::
    :label: 8.7-1

    \rho_{\text{f}} = \frac{\rho_{\text{o}}}{1 + C_{1} \left( T - 273 \right) + C_{2} \left( T - 273 \right)^{2}}

where

	:math:`\rho_{\text{o}}` = The theoretical density at 273 K, kg/m\ :sup:`3`

	:math:`C_{1},C_{2}` = Input coefficients

	:math:`T` = Temperature, K

This applies between 273 K and the solidus temperature.

The liquid fuel density is given by

.. math::
    :label: 8.7-2

    \rho_{\text{l}} = \frac{\rho_{\text{o}}}{1 + C_{3} \left( T - 273 \right)}

where

	:math:`C_{3}` = Input coefficient

This applies to temperatures above the liquidus. For the range between
the solidus and liquidus temperatures, a linear interpolation is
performed.

These equations are found in the function RHOF. Suggested values of
coefficients are from the Nuclear Systems Materials Handbook [8‑17].

	:math:`\rho_{\text{o}} = \text{COEFDS} \left( 1 \right) = 11.05 \times 10^{3}` kg/m\ :sup:`3` (mixed
	oxide)

	:math:`C_{1} = \text{COEFDS} \left( 2 \right) = 2.04 \times 10^{-5}` K\ :sup:`-1`

	:math:`C_{2} = \text{COEFDS} \left( 3 \right) = 8.70 \times 10^{-9}` K\ :sup:`-2`

	:math:`C_{3} = \text{COEFDL} \left( 2 \right) = 9.30 \times 10^{-5}` K\ :sup:`-1`

.. _section-8.7.2:

Fuel Thermal Expansion Coefficient
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The thermal expansion coefficient of the fuel is derived from the
coefficients of the fuel density function. This assures consistency
between the thermal expansion and the density.

The specific volume (inverse of density) at two temperatures is given by

.. math::
    :label: 8.7-3

    v_{1} = \frac{1 + C_{1} \left( T_{1} - 273 \right) + C_{2} \left( T_{1} - 273 \right)^{2}}{\rho_{\text{o}}}

.. math::
    :label: 8.7-4

    v_{2} = \frac{1 + C_{1} \left( T_{2} - 273 \right) + C_{2} \left( T_{2} - 273 \right)^{2}}{\rho_{\text{o}}}

The change in volume is therefore given by

.. math::
    :label: 8.7-5

    \Delta v_{1} = v_{2} - v_{1} = \frac{1}{\rho_{\text{o}}} \left\{ \left\lbrack C_{1} + C_{2} \left( T_{2} - 273 \right) \right\rbrack \left( T_{2} - 273 \right) \right. \\
    \left. - \left\lbrack C_{1} + C_{2} \left( T_{1} - 273 \right) \right\rbrack\left( T_{1} - 273 \right) \right\}

:eq:`8.7-5` can be rewritten in terms of the bulk thermal expansion
coefficient, :math:`\beta`.

.. math::
    :label: 8.7-6

    \Delta v = \frac{1}{\rho_{\text{o}}} \left\{ \beta \left( T_{2} \right) \left( T_{2} - T_{\text{r}} \right) - \beta \left( T_{1} \right) \left( T_{2} - T_{\text{r}} \right) \right\}

where

	:math:`\beta \left( T \right) = C_{1} + C_{2} T`

	:math:`T` = Temperature, K

	:math:`C_{1},C_{2}` = Density function coefficients

	:math:`T_{\text{r}}` = Reference temperature, K

Since the linear expansion coefficient is assumed to be a third of the
bulk expansion coefficient, the linear expansion coefficient can be
defined from the density coefficients as

.. math::
    :label: 8.7-7

    \alpha \left( T \right) + {C'}_{1} + {C'}_{2} T

where

	:math:`\alpha` = Linear thermal expansion coefficient

	:math:`{C'}_{1} = \frac{C_{1}}{3}`

	:math:`{C'}_{2} = \frac{C_{2}}{3}`

	:math:`T` = Temperature, K

This is found in the function ALPHF.

.. _section-8.7.3:

Fuel Modulus of Elasticity
~~~~~~~~~~~~~~~~~~~~~~~~~~

The modulus of elasticity, Young's Modulus, is determined from the bulk
modulus by

.. math::
    :label: 8.7-8

    E = 3\left( 1 - 2v \right) K

where

	:math:`E` = Modulus of elasticity, Pa

	:math:`K` = Bulk modulus, Pa

	:math:`v` = Poisson's ratio

The bulk modulus is from the NSMH [8‑17] and is given by

.. math::
    :label: 8.7-9

    K = 1.38 \times 10^{11} \left\lbrack 1 - 0.5 \left( T/2760 \right)^{2} \right\rbrack \left( 1 - 2P \right)

where

	:math:`K` = Bulk moduus, Pa

	:math:`T` = Temperature, °C

	:math:`P = 1 - \rho_{\text{TD}}` = Porosity fraction

	:math:`\rho_{\text{TD}}` = Fraction of theoretical density

Based on the evaluation in MATPRO‑l0 [8‑11], if this calculated modulus
of elasticity becomes less than :math:`7.9 \times 10^{8}`, it is fixed at this
value.

These equations are solved in the function EFUELF.

.. _section-8.7.4:

Mixed Oxide Fuel Thermal Conductivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Four different options exist for the fuel thermal conductivity. These
are controlled through the input parameter IRHOK.

**IRHOK = 0**

The thermal conductivity as a function of temperature is input in table
form through the variable arrays XKTAB and XKTEM.

**IRHOK = 1**

For this option, the conductivity equations [8-3] are given by

.. math::
    :label: 8.7-10

	\begin{aligned}
    k_{1} \left( T \right) = 1.1 + \frac{1 \times 10^{2}}{T \left( 0.4848 - 0.4465f_{\text{D}} \right)} && \text{for } 800 \text{ C} \leq T \leq 2000 \text{ C}
	\end{aligned}

.. math::
    :label: 8.7-11

	\begin{aligned}
    k_{1} \left( T \right) = k_{1} \left( 800 \right) \frac{168.844}{12.044 + \left( 0.196 \right)T} && \text{for } T < 800 \text{ C}
	\end{aligned}

.. math::
    :label: 8.7-12

	\begin{aligned}
    k_{3} \left( T \right) = k_{1} \left( 2000 \right) && \text{for } T > 2000 \text{ C}
	\end{aligned}

where

	:math:`k_{1}`, :math:`k_{2}`, :math:`k_{3}` = Fuel thermal conductivity, W/m-K

	:math:`T` = Temperature, °C

	:math:`f_{\text{D}}` = Fuel fraction of theoretical density

**IRHOK = 2**

This form of the conductivity [8‑22] is given by

.. math::
    :label: 8.7-13

	\begin{aligned}
    k_{1} \left( T \right) = \left\lbrack \left( C_{1} - f_{\text{D}} \right) f_{\text{D}} - 1 \right\rbrack\left\lbrack \frac{1}{\left( C_{1} + C_{3} T \right)} + C_{4} T^{3} \right\rbrack && \text{for } 0.75 \leq f_{\text{D}} \leq 0.95
	\end{aligned}

.. math::
    :label: 8.7-14

	\begin{aligned}
    k_{2} \left( T \right) = \left( 3f_{\text{D}} - 1 \right) \left\lbrack \frac{1}{\left( C_{5} + C_{6} T \right)} + C_{7} T^{3} \right\rbrack && \text{for } f_{\text{D}} > 0.95
	\end{aligned}

where

	:math:`C_{1}`, :math:`C_{2}`, :math:`C_{3}`, :math:`C_{4}`, :math:`C_{5}`, :math:`C_{6}`, :math:`C_{7}` = Input variables

	:math:`k_{\text{l}}`, :math:`k_{2}` = Fuel conductivity W/m-K

	:math:`T` = Temperature, K

If :math:`T` is greater than the melting temperature, it is set to the melting
temperature.

Suggested values:

	:math:`C_{1} = \text{COEFK} \left( 1 \right) = 2.1`

	:math:`C_{2} = \text{COEFK} \left( 2 \right) = 2.88 \times 10^{-3}`

	:math:`C_{3} = \text{COEFK} \left( 3 \right) = 2.52 \times 10^{-5}`

	:math:`C_{4} = \text{COEFK} \left( 4 \right) = 5.83 \times 10^{-10}`

	:math:`C_{5} = \text{COEFK} \left( 5 \right) = 5.75 \times 10^{-2}`

	:math:`C_{6} = \text{COEFK} \left( 6 \right) = 5.03 \times 10^{-4}`

	:math:`C_{7} = \text{COEFK} \left( 7 \right) = 2.91 \times 10^{-11}`

**IRHOK = 3**

This conductivity form is [8‑32]

.. math::
    :label: 8.7-15

	k_{1} \left( T \right) = \frac{4.005 \times 10^{3}}{\left( T - 273 \right) + 402.4} + 0.6416 \times 10^{-10} T^{3}

where

	:math:`T` = Temperature, K

	:math:`k` = Conductivity in W/m-K

This is the correlation for *UO*\ :sub:`2` and is converted to mixed
oxide by subtracting 0.2.

.. math::
    :label: 8.7-16

	k_{2} \left( T \right) = k_{1} \left( T \right) - 0.2

The porosity correction term was derived for use in the COMETHE‑IIIJ
[8‑33] code and is given by

.. math::
    :label: 8.7-17

	f_{\text{p}} = 1 - 1.029 \varepsilon - 3.2 \varepsilon^{2} - 40.1 \varepsilon^{3} + 158 \varepsilon^{4}

where

	:math:`f_{\text{p}}` = Porosity multiplier

	:math:`\varepsilon = 1 - \rho_{\text{f}}` = Fractional porosity

	:math:`\rho_{\text{f}}` = Fractional fuel density = actual density/theoretical
	density

The conductivity is therefore given by

.. math::
    :label: 8.7-18

	k \left( T \right) = f_{\text{p}} \times k_{2} \left( T \right)

Two different routines contain the above correlations, FK and KFUEL. The
function FK returns a single value of the conductivity for a single
invocation and is used in the pre‑transient. The subroutine KFUEL
returns the conductivity values for each radial node in the current
axial segment. It is used in the transient calculational procedure.

.. _section-8.7.5:

Mixed Oxide Fuel Fracture Strength
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The fracture strength for mixed‑oxide fuel is from the NSMH [8-17] and
is for unirradiated fuel.

.. math::
    :label: 8.7-19

	\sigma_{\text{f}} = 2.74 \times 10^{7} + 5.9 \times 10^{4} T

This is coded in the function SIGFRA.

.. _section-8.7.6:

Mixed Oxide Fuel Creep Rate
~~~~~~~~~~~~~~~~~~~~~~~~~~~

The fuel creep rate function is from the NSMH [8‑17] and is used in the
subroutine FSWELL. It represents diffusional flow, dislocation creep,
fission‑enhanced thermal creep, and fission‑induced creep.

.. math::
    :label: 8.7-20

    {\dot{\varepsilon}}_{\text{total}} = \frac{A}{d^{2}} \left\lbrack 1 + 2.11 \left( 97 - \text{TD} \right) \right\rbrack\sigma \exp\left( - \frac{Q_{1}}{\text{RT}} \right) \\
    + B \left\lbrack 1 + 0.22 \left( 97 - \text{TD} \right) \right\rbrack \sigma^{4.4} \exp\left( - \frac{Q_{2}}{\text{RT}} \right) \\
    + C \sigma \dot{F} \exp\left( - \frac{Q_{3}}{\text{RT}} \right) + D \sigma \dot{F}

where

	:math:`{\dot{\varepsilon}}_{\text{total}}` = Creep rate, s\ :sup:`-1`

	:math:`A = 8.97222 \times 10^{5}`

	:math:`d` = Grain size, :math:`\mu \text{m}`

	:math:`\text{TD}` = Fuel percent of theoretical density

	:math:`\sigma` = Stress, MPa

	:math:`Q_{1} = 3.87173 \times 10^{5}`

	:math:`R` = Gas constant = 8.3169 J/mole-K

	:math:`T` = Temperature, K

	:math:`\beta = 9.0 \times 10^{2}`

	:math:`Q_{2} = 5.72598 \times 10^{5}`

	:math:`C = 7.8889 \times 10^{-21}`

	:math:`F` = Fission rate, fissions/cm\ :sup:`3`\ -s

	:math:`Q_{3} = 5.7343 \times 10^{4}`

	:math:`D = 1.5 \times 10^{-24}`

This is used to calculate the fuel creep for fission‑gas bubble
expansion or contraction in the routine FSWELL.

.. _section-8.7.7:

Cladding Thermal Expansion Coefficient
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The mean thermal expansion coefficient of the cladding is taken from the
Nuclear Systems Materials Handbook [8‑17]. It has the form

.. math::
    :label: 8.7-21

	\alpha_{\text{m}} = C_{1} \left( C_{2} \right)^{\left( C_{3} \right)^{T}}

where

	:math:`\alpha_{\text{m}}` = Mean thermal expansion coefficient, °F\ :sup:`-1`

	:math:`C_{1}`, :math:`C_{2}`, :math:`C_{3}` = Calibration constants

	:math:`T` = Temperature, °F

The values for the calibration are given below:

	:math:`C_{1} = 11.397`

	:math:`C_{2} = 0.71828`

	:math:`C_{3} = 0.99890`

This is calculated in the function ALPHC.

.. _section-8.7.8:

Cladding Modulus of Elasticity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The cladding modulus of elasticity is taken from the NSHM [8-17] and is
given by

.. math::
    :label: 8.7-22

	E_{\text{c}} = 2.833669 \times 10^{1} - 2.882211 \times 10^{- 3} T - 3.687849 \times 10^{-6} T^{2} + 7.709188 \times 10^{-10} T^{3}

where

	:math:`E` = Modulus of elasticity, Mpsi

	:math:`T` = Temperature, °F

This is then converted to Pa by multiplying by :math:`6.894757 \times 10^{9}`.
This correlation is coded in the function ECLADF.

.. _section-8.7.9:

Cladding Ultimate Tensile Strength
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The ultimate tensile strength of unirradiated, 20% cold‑worked, 316 SS
is taken from the NSMH [8‑17].

.. math::
    :label: 8.7-23

	\sigma_{\text{u}} = C_{0} + C_{1} T + C_{2}T^{2} + C_{3}T^{3} + C_{4} T^{4} + C_{5} T^{5} + C_{6} T^{6} + C_{7} T^{7}

where

	:math:`\sigma_{\text{u}}` = Ultimate tensile strength, ksi

	:math:`T` = Temperature, F

	:math:`C_{0} = 1.220241 \times 10^{2}`

	:math:`C_{1} = -1.015998 \times 10^{-1}`

	:math:`C_{2} = 8.336636 \times 10^{-4}`

	:math:`C_{3} = -3.365737 \times 10^{-6}`

	:math:`C_{4} = 6.227377 \times 10^{-9}`

	:math:`C_{5} = -5.736229 \times 10^{-12}`

	:math:`C_{6} = 2.542064 \times 10^{-15}`

	:math:`C_{7} = -4.321098 \times 10^{-19}`

This is then converted to SI units by multiplying by
:math:`6.894757 \times 10^{6}`.

This correlation is good for temperatures below 1200 K only. For
temperatures above 1200 K, the curve is assumed to go to zero at the
cladding solidus according to the following equation.

.. math::
    :label: 8.7-24

	\sigma_{\text{u}} = \sigma_{\text{u}} \left( 1200K \right) \left\{ 1 - \left\lbrack \frac{T - 1200}{T_{\text{mc}} - 1200} \right\rbrack^{2} \right\}

where

	:math:`\sigma_{\text{u}} \left( 1200 K \right) = 1.122 \times 10^{8}` Pa

	:math:`T` = Temperature, K

	:math:`T_{\text{mc}}` = Cladding solidus temperature, K

Care should be exercised when using the function UTS since it is for
unirradiated cladding and may be inconsistent with the option used for
the cladding flow stress.

.. _section-8.7.10:

Cladding Flow Stress
~~~~~~~~~~~~~~~~~~~~

There exist four different options available to control the cladding
flow stress calculation in the function YLDCF. These are controlled
through the input parameter IYLD.

**IYLD = 0**

The value is calculated from the correlation in the NSMH [8‑17] for
unirradiated, 20% cold‑worked, 316 SS. For temperatures below 1200 K the
following form is used. It should be noted that this correlation is for
the strain rate of the tensile tests and may not be appropriate in many
accident cases with high strain rates.

.. math::
    :label: 8.7-25

	\sigma_{\text{y}} = C_{0} + C_{1} T + C_{2} T^{2} + C_{3} T^{3} + C_{4} T^{4} + C_{5} T^{5} + C_{6} T^{6} + C_{7} T^{7} + C_{8} T^{8} + C_{9} T^{9} + C_{10} T^{10}

where

	:math:`\sigma_{\text{y}}` = Clad flow stress, ksi

	:math:`T` = Temperature, °F

	:math:`C_{0} = 9.611825 \times 10^{1}`

	:math:`C_{1} = -1.262505 \times 10^{-1}`

	:math:`C_{2} = 1.510991 \times 10^{-3}`

	:math:`C_{3} = -1.021806 \times 10^{-5}`

	:math:`C_{4} = 3.796623 \times 10^{-8}`

	:math:`C_{5} = -8.438888 \times 10^{-11}`

	:math:`C_{6} = 1.163911 \times 10^{-13}`

	:math:`C_{7} = -9.993892 \times 10^{-17}`

	:math:`C_{8} = 5.177194 \times 10^{-20}`

	:math:`C_{9} = -1.477323 \times 10^{-23}`

	:math:`C_{10} = 1.780710 \times 10^{-27}`

The value is then converted to Pa by multiplying by
:math:`6.894757 \times 10^{6}`.

For temperatures above 1200 K, a form like :eq:`8.7-24` is used for the
extrapolation to the melting temperature.

.. math::
    :label: 8.7-26

	\sigma_{\text{y}} = \sigma_{\text{y}} \left( 1200 \right) \left\{ 1 - \left\lbrack \frac{T - 1200}{T_{\text{mc}} - 1200} \right\rbrack^{2} \right\}

where

	:math:`\sigma_{\text{y}} \left( 1200 \right) = 7.375 \times 10^{7}` Pa

	:math:`T` = Temperature, K

	:math:`T_{\text{mc}}` = Clad solidus temperature, K

**IYLD = 1,2**

For these values of IYLD, a flow stress model developed by DiMelfi and
Kramer [8‑34] is used. If IYLD = 1, the model is temperature, strain,
strain rate, and burnup dependent. If IYLD = 2, a high strain rate
approximation is used, removing the strain rate dependence.

The flow stress, :math:`\sigma`, is defined by

.. math::
    :label: 8.7-27

	\sigma = \sigma_{\text{s}} - \left( \sigma_{\text{s}} - \sigma_{1} \right) \exp\left( - \frac{\hat{\varepsilon}}{\varepsilon_{\text{c}}} \right)

where

	:math:`\sigma_{1}` = Yield stress of fully annealed, unirradiated material

	:math:`\sigma_{\text{S}}` = Saturation flow stress approached as increases

	:math:`\hat{\varepsilon}` = Hardness parameter

	:math:`\varepsilon_{\text{c}}` = Material parameter

the hardness parameter :math:`\hat{\varepsilon}` has two components;
:math:`\varepsilon_{\text{p}}`, the accumulated equivalent plastic strain, and
:math:`{\hat{\varepsilon}}_{\phi}` the hardness due to irradiation,
which is determined from

.. math::
    :label: 8.7-28

	{\hat{\varepsilon}}_{\phi} = - \varepsilon_{\text{c}} \ln\left( 1 - \left\{ 1 - \exp\left\lbrack - B \left( \phi t - \phi_{\text{o}} t_{\text{o}} \right) \right\rbrack \right\}^{1/2} \right)

where

	:math:`B = 3.5 \times 10^{-27}`, m\ :sup:`2`/n

	:math:`\phi_{\text{o}} t_{\text{o}} = 4.5 \times 10^{25}`, n/m\ :sup:`2`

	:math:`\phi t` = Neutron fluence, n/m\ :sup:`2`

and

.. math::
    :label: 8.7-29

	\dot{\varepsilon} = \varepsilon_{\text{p}} + {\hat{\varepsilon}}_{\phi} + \varepsilon_{\text{cw}}

where

	:math:`\varepsilon_{\text{cw}}` = As-fabricated cold‑work strain

The relationship between :math:`\sigma_{\text{s}}`, :math:`\sigma_{1}`, and :math:`\varepsilon_{\text{c}}` is
given by

.. math::
    :label: 8.7-30

	\left\lbrack \frac{\sigma_{\text{s}}}{G} - \frac{\sigma_{1}}{G} \right\rbrack \frac{1}{\varepsilon_{\text{c}}} = \frac{\theta_{1}}{G}

where

	:math:`\theta_{1}/G = 3.66 \times 10^{-2}`

	:math:`G = 92.0 - 4.02 \times 10^{-2}` T, GPa

This relationship is equivalent to assuming that the initial
work‑hardening rate :math:`\theta_{1}`, for annealed material
:math:`\left( \hat{\varepsilon} = 0 \right)` is constant.

The functions :math:`\sigma_{\text{s}}` and :math:`\sigma_{1}` are chosen to be of the
form

.. math::
    :label: 8.7-31

	\frac{\sigma_{\text{s}}}{G} = \frac{\sigma_{\text{SO}}}{G} \left\{ 1 - \exp\left\lbrack - \left( \frac{\dot{\varepsilon}_{\text{p}}}{\dot{\varepsilon}_{\text{OS}}} \right)^{nk} \right\rbrack \right\}^{1/k}

.. math::
    :label: 8.7-32

	\frac{\sigma_{1}}{G} = \frac{\sigma_{10}}{G} \left\{ 1 - \exp\left\lbrack - \left( \frac{\dot{\varepsilon}_{\text{p}}}{\dot{\varepsilon}_{01}} \right)^{nk} \right\rbrack \right\}^{1/k}

where

	:math:`{\hat{\varepsilon}}_{\text{p}}` = Equivalent plastic strain rate

	n = Constant = 1/5.35

	k = Constant = 2.0

The functions :math:`\sigma_{\text{SO}}`, :math:`\sigma_{10}`,
:math:`{\dot{\varepsilon}}_{\text{OS}}`, :math:`{\dot{\varepsilon}}_{01}`
are all temperature dependent. The
constant :math:`k` is a nonphysical parameter that governs the sharpness of
the transition between strain‑rate‑dependent and strain‑rate‑independent
behavior.

If IYLD = 2, then the above relationships for :math:`\sigma_{\text{s}}` and
:math:`\sigma_{1}` are assumed to have the form

.. math::
    :label: 8.7-33

	\frac{\sigma_{\text{s}}}{G} = \frac{\sigma_{\text{SO}}}{G}

and

.. math::
    :label: 8.7-34

	\frac{\sigma_{1}}{G} = \frac{\sigma_{10}}{G}

In both cases, IYLD = 1 or IYLD = 2, the following functions are used

.. math::
    :label: 8.7-35

	\frac{\sigma_{\text{SO}}}{G} = 2.06 \times 10^{-3} + \frac{7.12 \times 10^{-1}}{T}

.. math::
    :label: 8.7-36

	\frac{\sigma_{10}}{G} = 2.00 \times 10^{-2} - 9.12 \times 10^{-6} T

The functions :math:`{\hat{\varepsilon}}_{\text{OS}}` and
:math:`{\hat{\varepsilon}}_{01}` are strong functions of temperature and
are related to the dominant creep mechanism. This is assumed to be
thermally activated, so Arrhenius relationships are used.

.. math::
    :label: 8.7-37

	{\dot{\varepsilon}}_{\text{OS}} = {\dot{\varepsilon}}_{\text{OOS}} \exp\left( - \frac{Q}{RT} \right)

.. math::
    :label: 8.7-38

	{\dot{\varepsilon}}_{01} = {\dot{\varepsilon}}_{001} \exp\left( - \frac{Q}{RT} \right)

where

	:math:`Q/R` = 38,533, K

	:math:`{\dot{\varepsilon}}_{\text{OOS}} = 1.062 \times 10^{14}`,
	s\ :sup:`-1`

	:math:`{\dot{\varepsilon}}_{001} = 3.794 \times 10^{12}`,
	s\ :sup:`-1`

Given the above constants and relationships, it is then possible to
calculate the flow stress of the cladding.

**IYLD = 3**

With this selection, the input flow stress table YLDTAB with its
corresponding temperature table YLDTEM are searched to find the flow
stress as a function of temperature. One table exists for each cladding
type.

.. _section-8.7.11:

Thermal Conductivity of Helium
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The thermal conductivity of the helium in the fuel pin free volume is
temperature dependent and has been fit to the following equation [8‑35].

.. math::
    :label: 8.7-39

	K = 1.43 \times 10^{-1} + 3.17 \times 10^{-4} \left( T - 273 \right) - 2.24 \times 10^{-8} \left( T - 273 \right)^{2}

where

	:math:`K` = Thermal conductivity, W m\ :sup:`-1` K\ :sup:`-1`

	:math:`T` = Gas temperature, K

This is used in the function routine HGAP.

.. _section-8.7.12:

Thermal Conductivity of Fission Gas
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

All fission gas is assumed to be xenon. A functional form of the xenon
conductivity [8‑35] is used in the function routine HGAP.

.. math::
    :label: 8.7-40

	K = 5.15 \times 10^{-3} + 1.69 \times 10^{-5} \left( T - 273 \right) - 3.50 \times 10^{-9} \left( T - 273 \right)^{2}

where

	:math:`K` = Thermal conductivity, W m\ :sup:`-1` K\ :sup:`-1`

	:math:`T` = Gas temperature, K

.. _section-8.7.13:

Fuel Hardness
~~~~~~~~~~~~~

In DEFORM‑4 it is assumed that conditions could arise where either the
fuel or cladding could be the softer material when considering
fuel‑cladding contact in the determination of the gap conductance. The
following equation is therefore used to determine the oxide fuel
hardness [8‑31].

.. math::
    :label: 8.7-41

	H = 6.009 \times 10^{9} \exp\left( - \frac{T}{641} \right)

where

	:math:`H` = Meyer hardness, Pa

	:math:`T` = Temperature, K

.. _section-8.7.14:

Cladding Hardness
~~~~~~~~~~~~~~~~~

The cladding hardness is calculated from the following equation [8‑21]
when comparing with the fuel hardness. The softer of the fuel or
cladding is used in the solid‑to‑solid gap conductance considerations.

.. math::
    :label: 8.7-42

	\begin{aligned}
	H = 5.961 \times 10^{9} T^{0.206}~, && T < 893.92 \text{ K}
	\end{aligned}

.. math::
    :label: 8.7-43

	\begin{aligned}
	H = 2.75 \times 10^{22} T^{6.53}~, && T > 893.92 \text{ K}
	\end{aligned}

where

	:math:`H` = Meyer hardness, Pa

	:math:`T` = Temperature, K

.. _section-8.7.15:

Cladding Fast Creep
~~~~~~~~~~~~~~~~~~~

Kramer and DiMelfi [8‑34] have shown that the following equation is
suitable for plastic strain of cladding at high temperatures.

.. math::
    :label: 8.7-44

	\dot{\varepsilon} = \frac{{\dot{\varepsilon}}_{\text{OOS}}}{\sigma_{\text{SO}}^{n}} \sigma^{n} \exp\left( - \frac{Q_{\text{c}}}{T} \right)

where

	:math:`\dot{\varepsilon}` = Plastic strain rate, s\ :sup:`-1`

	:math:`\sigma` = Equivalent stress in the cladding, Pa

	:math:`T` = Temperature, K

	:math:`{\dot{\varepsilon}}_{\text{OOS}} = 1.062 \times 10^{14}`,
	s\ :sup:`-1`

	:math:`\sigma_{\text{so}}` = Material constant defined in :eq:`8.7-35`

	:math:`n` = Stress exponent = 5.35

	:math:`Q_{\text{c}}` = Creep activation constant = 38,533 K

This allows for plastic deformation of the cladding below the flow
stress.

.. _section-8.7.16:

Material Properties of Metal Alloy Fuel
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Material properties for the metal fuels under consideration for the
Integral Fast Reactor concept are currently under investigation in order
to prepare a set of recommended values for use by the safety community.
Unlike the oxide fuel which has been under study for a number of years,
the metal fuel data is fairly sparse and limited to specific fuel types.
Therefore, the correlations given below are to be considered preliminary
in nature, but they should provide results that are consistent with
future results. These have been coded into DEFORM in a manner that makes
future modifications a simple replacement task.

Several thermal properties are also used by DEFORM and these have been
addressed in :numref:`Chapter %s<section-10>` of this document. Most of the properties given
below are obtained from a survey of data carried out for the
modification of the FPIN code to analyze metal fuels [8-36].

.. _section-8.7.16.1:

Modulus of Elasticity
^^^^^^^^^^^^^^^^^^^^^

Although the modulus of elasticity varies with temperature, composition,
and phase, a single average value has been used.

.. math::
    :label: 8.7-45

	E_{\text{f}} = 1.4 \times 10^{12}

where

	:math:`E_{\text{f}}` = Modulus of elasticity, Pa

.. _section-8.7.16.2:

Poisson's Ratio
^^^^^^^^^^^^^^^

Poisson's ratio is an input parameter to the SAS4A code. For both the
ternary and fission alloy fuels the currently recommended value is 0.23.

.. _section-8.7.16.3:

Fracture Strength
^^^^^^^^^^^^^^^^^

Unlike the oxide fuels, the metal fuel does not exhibit a fracture
phenomenon. While there does appear to be a separation between the
phases that result in the ternary fuels, there is no radial cracking.
This appears to be the result of the softness of the metal matrix and
the inherent structural differences that exist between ceramics and
metals. When metal fuel has been specified, the function SIGFRA returns
a large value for the fracture strength to avoid cracking during the
mechanical calculations.

.. _section-8.7.16.4:

Fuel Creep Rate
^^^^^^^^^^^^^^^

One of the more important correlations used by DEFORM is the creep rate
of the fuel because of its influence on the rate of fuel swelling
through the process of fission gas bubble volume change. For the metal
fuels the creep rate, determined in a study performed by John Kramer
[8-37], has been employed for both types of metal fuel currently
considered in DEFORM.

For uranium phases existing below 973 K, the rate is dependent on two
stress related terms.

.. math::
    :label: 8.7-46

	R_{\text{c}} = \left( 0.5 \times 10^{4} \sigma + 6.0 \sigma^{4.5} \right) \exp\left( - \frac{26168}{T} \right)

where

	:math:`R_{\text{c}}` = Fuel creep rate, s\ :sup:`-1`

	:math:`\Sigma` = Stress, MPa

	T = Temperature, K

For the high temperature gamma uranium phase, the rate is give by a
single stress dependent function.

.. math::
    :label: 8.7-47

	R_{\text{c}} = 8.0 \times 10^{-2} \sigma^{3} \exp\left( - \frac{14353}{T} \right)

The time constant for bubble expansion is the inverse of the calculated
creep rate. These functions are coded into the subroutine FSWELL.

.. _ti-cladding-properties:

15-15Ti Cladding Properties
~~~~~~~~~~~~~~~~~~~~~~~~~~~

Thermal Creep
^^^^^^^^^^^^^

Given the cladding equivalent stress and clad mid-wall temperature at
each axial location, the thermal creep correlation computes the thermal
creep strain rate at each time step during normal operation as well as
transients. The thermal creep equation derived in [8.39] for 15-15Ti
cladding has been adopted as is. The form of the equation is given as
follows:

.. math::
    :label: 1515TiThermalCreep

    {\dot{\epsilon}}_{th,\ Eq} = \left\lbrack sinh(\beta\sigma_{\text{Eq}}) \right\rbrack^{n}e^{- \frac{E}{\text{RT}}}

.. math:: E = E_{1} + E_{2}T + E_{3}T^{2}

Where :math:`{\dot{\epsilon}}_{th,Eq}` is the thermal creep strain rate
(1/s), :math:`\sigma_{\text{Eq}}` is the equivalent stress (MPa), and
:math:`T` is the clad mid-wall temperature (K). :math:`R` is the gas
constant, 8.3144621 J/mol/K, and :math:`E` is the activation energy
(J/mol). :math:`\beta`, :math:`n`, and :math:`E_{1}`,\ :math:`\ E_{2}`,
and :math:`E_{3}` are the fitting parameters, which are given in :numref:`tab1515TiCreepParameters` below.

.. _tab1515TiCreepParameters:

.. table:: Fitting parameters for the 15-15Ti Thermal Creep Model
	:align: center
	:widths: 1 1

	===================== =========
	Fitting Parameter     Value
	===================== =========
	:math:`\beta` (1/MPa) 1.4994E-2
	:math:`E_{1}` (J/mol) -6.591E+5
	:math:`E_{2}` (J/mol) 1.8211E+3
	:math:`E_{3}` (J/mol) -1.0513
	:math:`n`             2.3319
	===================== =========

Irradiation Creep
^^^^^^^^^^^^^^^^^

Given the cladding hoop stress, clad mid-wall temperature, and clad dose
rate, which is a function of clad fast neutron flux, at each axial
location, the irradiation creep correlation computes the irradiation
creep strain rate at each time step. The irradiation creep equation
derived in [8.40] for 15-15Ti cladding has been adopted as is. The form
of the equation is given as follows:

.. math::
    :label: 1515TiIrradiationCreep

    {\dot{\epsilon}}_{ir,\ \theta} = 0.75 \times A \times \sigma_{\theta} \times {\dot{\varphi}}_{d}

Where :math:`{\dot{\epsilon}}_{ir,\ \theta}` is the irradiation creep
strain rate (1/s), :math:`\sigma_{\theta}` is the clad hoop stress
(MPa), and :math:`{\dot{\varphi}}_{d}` is the clad dose rate (dpa/s).
The cladding dose rate is computed by multiplying the fast neutron flux
with the dose conversion factor, :sasinp:`CDOSECONV`. The fast neutron flux is computed by multiplying
:sasinp:`FLTPOW` with the linear heat rate.
:math:`A` is the fitting parameter given as a function of clad mid-wall
temperature, which is shown in :numref:`fig1515TiCreepModulus`. Below 673 K and above 898 K,
the value of :math:`A` is set to its value at 673 K and 898 K,
respectively.

.. _fig1515TiCreepModulus:

.. figure:: media/fig1515TiCreepModulus.png
    :align: center
    :figclass: align-center

    Irradiation Creep Modulus for 15-15Ti Cladding

Void Swelling
^^^^^^^^^^^^^

Void swelling rate of cladding is modeled as a function of cladding dose
and cladding dose rate, which is given as a function of fast neutron
flux and temperature.
The cladding dose rate is computed by multiplying the fast neutron flux
with the dose conversion factor, :sasinp:`CDOSECONV`. The fast neutron flux is computed by multiplying
:sasinp:`FLTPOW` with the linear heat rate.
The void swelling plots given in :numref:`fig1515TiCladSwelling` [8.40]
as a function of cladding dose and clad temperature are used to derive
linear piecewise continuous functions to predict the void swelling rate
at a given time step.

.. _fig1515TiCladSwelling:

.. figure:: media/fig1515TiCladSwelling.png
    :align: center
    :figclass: align-center

    Experimental Data for volumetric clad swelling as a function
    of dose at 733 K, 773 K and 813 K for 15-15Ti cladding

Transient Creep Rupture
^^^^^^^^^^^^^^^^^^^^^^^

The creep rupture margin of the cladding is computed using the CDF/life
fraction model and is a function of hoop stress, mid-wall temperature
and dose of the cladding. CDF at any time, :math:`t_{0}`, during the
transient is described by the following relation:

.. math::
    :label: 1515TiCDFLifeFraction

    \text{CDF}\left( t_{0} \right) = \int_{0}^{t_{0}}\frac{\text{dt}}{t_{r}(\sigma_{\text{eq}},T,\varphi_{d})}

where, :math:`t_{r}` is the time to rupture (s),
:math:`\sigma_{\text{eq}}` is the equivalent stress (MPa), :math:`T` is
the temperature (K), and :math:`\varphi_{d}` is the dose (dpa).

Time to rupture correlations given in [8.39] for the unirradiated
15-15Ti have been extended in order to account for the neutron
irradiation induced damage and adopted in this model. The following form
is used:

.. math::
    :label: 1515TiTimeToRupture

    P = \begin{cases}\left\lbrack \frac{A_{1}}{A_{3}}\left( \log\sigma_{\text{eq}} - A_{2} \right) + 1 \right\rbrack\left\lbrack 1 + A_{3} - e^{A_{4}\left( \log\sigma_{\text{eq}} - A_{2} \right)} \right\rbrack & T < 1023.15 \\
    B_{1}\log\sigma_{\text{eq}} + B_{2} & T \geq 1023.15
    \end{cases}

.. math:: t_{r} = i_{f} \times 10^{\left( P\frac{1000}{T} - 17.6 \right)} \times 3600

where, :math:`P` is the Larson-Miller parameter and :math:`A_{1}`,
:math:`A_{2}`, :math:`A_{3}`, :math:`A_{4}`, :math:`B_{1}`, and
:math:`B_{2}` are the fitting parameters given in :numref:`tab1515TiRuptureParameters`.

.. _tab1515TiRuptureParameters:

.. table:: Fitting parameters for 15-15Ti Time to rupture correlation
	:align: center
	:widths: 1 1

	================= =======
	Fitting Parameter Value
	================= =======
	:math:`A_{1}`     -0.78
	:math:`A_{2}`     2.3054
	:math:`A_{3}`     19.512
	:math:`A_{4}`     5.196
	:math:`B_{1}`     -6.1175
	:math:`B_{2}`     31.7882
	================= =======

The modeling
of the irradiation induced embrittlement of 15-15Ti was approximated
using clad void swelling data and the interrelation between void
swelling and clad embrittlement for similar stainless steels such as
SS316 and D9 using the creep rupture data given in Refs. [8.41], [8.42],
and [8.43]. The :math:`\ i_{f}` correction factor is modeled as a function of
15-15-Ti cladding dose as given in :numref:`fig1515TiCorrectionFactor`.
The cladding dose rate is computed by multiplying the fast neutron flux
with the dose conversion factor, :sasinp:`CDOSECONV`. The fast neutron flux is computed by multiplying
:sasinp:`FLTPOW` with the linear heat rate.
At high doses where void
swelling takes off exceeds 5% and embrittles cladding, the irradiation
factor is assumed to be 0.01 consistent with the findings for SS316
cladding. This value is also used for cladding doses that are beyond 120
dpa.

.. _fig1515TiCorrectionFactor:

.. figure:: media/fig1515TiCorrectionFactor.png
	:align: center
	:figclass: align-center

	Approximate irradiation correction factor for 15-15Ti time to rupture as a function of cladding dose

In order to model the stochastic nature of clad failure, a normal
probability density function (PDF) has been adopted as a function
of the logarithm of the CDF.

.. math::
    :label: 1515TiStochasticRupture

    \mathrm{\text{PDF}} = \frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left\lbrack - \frac{\left( \log\mathrm{\text{CDF}} - \mu \right)^{2}}{2\sigma^{2}} \right\rbrack

where :math:`\sigma` is the standard deviation and :math:`\mu` is the
mean in units of the logarithm of CDF.

The integral of the PDF given by the equation above is interpreted as
the fraction of the fuel pins that are failed. PDF and integral of PDF
for :math:`\sigma = 1` and :math:`\mu = 0` are plotted in :numref:`fig1515TiPDF`. The
model currently ignores potential pin-to-pin failure propagation due to
fission gas and fuel ejection to the channel, clad ballooning, and
blockages.

.. _fig1515TiPDF:

.. figure:: media/fig1515TiPDF.png
	:align: center
	:figclass: align-center

	An example PDF and integral of PDF for :math:`\sigma=1` and :math:`\mu=0`

.. _CladdingOuterCorrosion:

Cladding Outer Corrosion
^^^^^^^^^^^^^^^^^^^^^^^^

Chemical interactions between stainless steel clad and lead coolant is a
critical issue of LFR designs. The highly complex nature of the problem
and unavailability of corrosion data led to the decision to keep this
model simple and mostly dependent on user input. The user is allowed to
define the thickness of the cladding wastage prior to the transient. The
corrosive layer is increased from zero to the set value during the
pre-transient characterization and is kept as constant during the
transient. The corroded outer layer of the cladding is assumed not to
bear any applied load.