.. _section-8.2:

Fuel-pin Mechanics
------------------

In the treatment adopted in DEFORM‑4, the fuel pin in an axial segment
is divided into 6 radial zones, not all of which need exist. These are
(1) the central void, (2) the molten fuel zone, (3) the solid,
continuous fuel zone, (4) the cracked fuel zone, (5) the fuel‑cladding
gap, and (6) the fuel‑pin cladding. The zones are illustrated in :numref:`figure-8.2-1`. Each zone may consist of one or more cells. These zones will be
explained in detail in the following sections.

The approach used is to divide the calculation into the thermoelastic
solution, and then superimpose on this the plastic deformation resulting
from fuel swelling or cladding stress induced plastic creep and
irradiation swelling

.. math::
    :label: 8.2-1

	\varepsilon_{\text{T}} = \varepsilon_{\text{e}} + \varepsilon_{\text{th}} + \varepsilon_{\text{s}}

where

:math:`\epsilon_{\text{r}}` = Total strain at the cell boundary

:math:`\epsilon_{\text{e}}` = Elastic strain from applied boundary forces

:math:`\epsilon_{\text{th}}` = Thermal expansion induced strain

:math:`\epsilon_{\text{s}}` = Swelling strains from the solid and volatile
fission products in the fuel, or irradiation induced void formation and
stress induced plastic creep in the cladding.

These terms are discussed in the following sections.

.. _figure-8.2-1:

..  figure:: media/image7.png
	:align: center
	:figclass: align-center
	:width: 6.19097in
	:height: 7.93264in

	DEFORM‑4 Radial Zones

The cladding is assumed to be an elastic‑perfectly‑plastic material.
Several functions are available as options to provide the flow stress of
the cladding. Once the fuel‑cladding interface pressure produces a
circumferential stress exceeding the flow stress, the interface pressure
is limited to that necessary to achieve the flow stress and the cladding
will follow the fuel deformation until the conditions bring the cladding
back into the elastic behavior region. One of the options for flow
stress includes the effects of previous strain, strain rate,
temperature, and irradiation history. When this option is used, the flow
stress changes as these parameters change, providing a work hardening
and strain‑rate dependence. Besides this perfectly plastic behavior,
there is plastic creep of the cladding at conditions below the flow
stress. This strain is calculated and added to the accumulated strain.

.. _section-8.2.1:

Solid Fuel and Cladding Thermoelastic Solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the solid fuel and cladding, the material is assumed to be
continuous, isotropic, elastic, and axisymmetric. Because of the
axisymmetry, all shear stresses and strains are assumed to be zero. The
generalization of Hooke's Law to three dimensions is used to provide the
linear elastic relationship between the stresses and strains. The
thermal expansion strains are included through the principle of
superposition of linear equations. These considerations therefore yield
the following set of constitutive equations.

.. math::
    :label: 8.2-2

	\varepsilon_{\text{r}} = \frac{1}{E} \left\lbrack \sigma_{\text{r}} - \nu \left( \sigma_{\theta} + \sigma_{\text{z}} \right) \right\rbrack + \Delta\left( \alpha T \right)

.. math::
    :label: 8.2-3

	\varepsilon_{\theta} = \frac{1}{E} \left\lbrack \sigma_{\theta} - \nu \left( \sigma_{\text{r}} + \sigma_{\text{z}} \right) \right\rbrack + \Delta\left( \alpha T \right)

.. math::
    :label: 8.2-4

	\varepsilon_{\text{z}} = \frac{1}{E} \left\lbrack \sigma_{\text{z}} - \nu \left( \sigma_{\text{r}} + \sigma_{\theta} \right) \right\rbrack + \Delta\left( \alpha T \right)

.. math::
    :label: 8.2-5

	\Delta\left( \alpha T \right) = \alpha \left( T_{2} \right) \left( T_{2} - T_{\text{r}} \right) - \alpha \left( T_{1} \right) \left( T_{1} - T_{\text{r}} \right)

where

:math:`\varepsilon_{\text{r}}`, :math:`\varepsilon_{\theta}`, :math:`\varepsilon_{\text{z}}` =
Strain in the radial, circumferential, and axial directions,
respectively

:math:`\sigma_{\text{r}}`, :math:`\sigma_{\theta}`, :math:`\sigma_{\text{z}}` = Stress in the
radial, circumferential, and axial directions, respectively

:math:`\upsilon` = Poisson's ratio for the material

:math:`\alpha(T)` = Mean linear thermal expansion coefficient

:math:`E` = Modulus of elasticity

:math:`T_{2}`, :math:`T_{1}` = Temperature at the final and initial states,
respectively,

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

The strains are related to the displacements through geometrical
considerations. In the cylindrical coordinate system used in SAS4A,
these kinematic equations are as follows:

.. math::
    :label: 8.2-6

	\varepsilon_{\text{r}} = \frac{\text{du}}{\text{dr}}

.. math::
    :label: 8.2-7

	\varepsilon_{\theta} = \frac{u}{r} + \frac{1}{r} \frac{\text{dv}}{\text{d}\theta}

.. math::
    :label: 8.2-8

	\varepsilon_{\text{z}} = \frac{\text{dw}}{\text{dz}}

where

:math:`u` = Displacement in the radial direction r

:math:`\upsilon` = Displacement in the circumferential direction θ

:math:`w` = Displacement in the axial direction z

Since material is assumed to be axisymmetric, there is no variation of v
circumferentially, so :eq:`8.2-7` reduces to

.. math::
    :label: 8.2-9

	\varepsilon_{\theta} = \frac{u}{r}

In order to be able to obtain simple analytical solutions with the above
equations, a generalized plane strain approximation has been employed.
Each axial segment is assumed to elongate uniformly over the cross
section to maintain a plane interface between segments. :eq:`8.2-8`
can therefore be rewritten as

.. math::
    :label: 8.2-10

	\varepsilon_{z} = z_{\text{o}}

where

:math:`z_{\text{o}}` = Axial plane strain for the segment

Since the cells under consideration are assumed to be at rest, with no
shear stresses, mechanical considerations provide the following equation
of equilibrium.

.. math::
    :label: 8.2-11

	\frac{\text{d}\sigma_{\text{r}}}{\text{dr}} + \frac{\sigma_{\text{r}} - \sigma_{\theta}}{r} = 0

:eq:`8.2-2` through :eq:`8.2-6`, and :eq:`8.2-9` through :eq:`8.2-11` form the set of
equations solved by DEFORM‑4 for the thermoelastic response.
:eq:`8.2-10` is substituted into :eq:`8.2-4` and this is solved for the axial
stress, :math:`\sigma_{\text{z}}`. This result is then substituted along with
:eq:`8.2-6` and :eq:`8.2-9` in :eq:`8.2-2` and :eq:`8.2-3` to yield

.. math::
    :label: 8.2-12

	\sigma_{\text{r}} = \frac{E}{\left( 1 + \nu \right) \left( 1 - 2\nu \right)} \left\lbrack \left( 1 - \nu \right) \frac{\text{du}}{\text{dr}} + \nu \frac{u}{r} + \nu z_{\text{o}} - \left( 1 + \nu \right) \Delta \left( \alpha T \right) \right\rbrack

.. math::
    :label: 8.2-13

	\sigma_{\theta} = \frac{E}{\left( 1 + \nu \right) \left( 1 - 2\nu \right)} \left\lbrack \nu \frac{\text{du}}{\text{dr}} + \left( 1 - \nu \right) \frac{u}{r} + \nu z_{\text{o}} - \left( 1 + \nu \right) \Delta \left( \alpha T \right) \right\rbrack

The stresses are expressed in terms of the radial displacement function
:math:`u \left( r \right)`. When these equations are used in the equilibrium :eq:`8.2-11`,
the following is obtained, assuming the modulus of elasticity, :math:`E`, is
constant over the region of interest. The value for the modulus of
elasticity is the mass‑weighted average of all those cells in this zone.

.. math::
    :label: 8.2-14

	\frac{1}{r} \frac{\text{d}}{\text{dr}} \left( r \frac{\text{du}}{\text{dr}} \right) - \frac{u}{r^{2}} = \frac{\left( 1 + \nu \right)}{\left( 1 - \nu \right)} \frac{\text{d}}{\text{dr}} \left\lbrack \Delta\left( \alpha T \right) \right\rbrack

The solution to this differential equation may be obtained as

.. math::
    :label: 8.2-15

	u\left( r \right) = \left( \frac{1 + \nu}{1 - \nu} \right)r I \left( r \right) + C_{1} r + \frac{C_{2}}{r}

where

:math:`C_{1},\ C_{2}` = Constants of integration

The function :math:`I \left( r \right)` is defined as

.. math::
    :label: 8.2-16

	I\left( r \right) = \frac{1}{r^{2}} \int_{\rho}^{r}{\Delta\left( \alpha T \right) r'} \text{dr}'

where

:math:`\rho` = Inner radius of the zone under consideration

If :eq:`8.2-15` is used to rewrite :eq:`8.2-12` and :eq:`8.2-13`, the radial and
circumferential stresses as functions of :math:`r`, :math:`C_{1}`, and :math:`C_{2}`
may be obtained.

.. math::
    :label: 8.2-17

	\frac{\sigma_{\text{r}}\left( r \right)}{E} = - \frac{I\left( r \right)}{\left( 1 - \nu \right)} + \frac{1}{\left( 1 + \nu \right) \left( 1 - 2\nu \right)} \left\lbrack C_{1} - \frac{\left( 1 - 2\nu \right) C_{2}}{r^{2}} + \nu z_{0} \right\rbrack

.. math::
    :label: 8.2-18

	\frac{\sigma_{\theta}\left( r \right)}{E} = \frac{I\left( r \right)}{\left( 1 - \nu \right)} + \frac{1}{\left( 1 + \nu \right) \left( 1 - 2\nu \right)} \left\lbrack C_{1} - \frac{\left( 1 - 2\nu \right) C_{2}}{r^{2}} + \nu z_{0} \right\rbrack - \left( \frac{\Delta \alpha T}{\left( 1 - \nu \right)} \right)

The constants of integration, :math:`C_{1}` and :math:`C_{2}`, may be
determined by the following boundary conditions:

.. math::
    :label: 8.2-19

	\sigma_{\text{r}} \left( r = \rho \right) = \sigma_{\rho}

.. math::
    :label: 8.2-20

	\sigma_{\text{r}} \left( r = \eta \right) = \sigma_{\eta}

where

:math:`\eta` = Outer radius of the zone under consideration

:math:`\sigma_{\rho}` = Stress at the inner surface of the zone

:math:`\sigma_{\eta}` = Stress at the outer surface of the zone

For the fuel, :math:`\rho` and :math:`\eta` would correspond to the inner solid and outer
uncracked cell boundaries, respectively. For the cladding, they would
correspond to the inner and outer surfaces, respectively.

The constants may then be determined by substituting :eq:`8.2-19` and
:eq:`8.2-20` into :eq:`8.2-17` and :eq:`8.2-18` and solving them simultaneously for
:math:`C_{1}` and :math:`C_{2}`.

.. math::
    :label: 8.2-21

	C_{1} = \frac{\left( 1 + v \right) \left( 1 - 2v \right)}{\left( \eta^{2} - \rho^{2} \right)} \left\lbrack \frac{I\left( \eta \right)\eta^{2}}{\left( 1 - v \right)} - \frac{1}{E}\rho\left( \rho^{2}\sigma_{\rho} - \eta^{2}\sigma_{\eta} \right) \right\rbrack - vz_{\text{o}}

.. math::
    :label: 8.2-22

	C_{2} = \frac{\left( 1 + \nu \right) \rho^{2}\eta^{2}}{\left( \eta^{2} - \rho^{2} \right)}\left\lbrack \frac{I\left( \eta \right)}{\left( 1 - \nu \right)} - \frac{\left( \sigma_{\rho} - \sigma_{\eta} \right)}{E} \right\rbrack

These constants may then be used in :eq:`8.2-15`, :eq:`8.2-17`, and :eq:`8.2-18` to
determine the radial displacement and stresses due to externally applied
forces and thermal expansion.

.. math::
    :label: 8.2-23

	u\left( r \right) = u_{\text{f}} \left( r \right) + u_{\text{th}} \left( r \right)

.. math::
    :label: 8.2-24

	u_{\text{f}}\left( r \right) &= \text{ Elastic displacement due to externally applied stress }\sigma_{\rho}\text{ and }\sigma_{\eta} \\
    \ &= \frac{\left( 1 + \nu \right)}{\left( \eta^{2} - \rho^{2} \right) E} \left\{ \eta^{2} \sigma_{\eta} \left\lbrack \left( 1 - 2\nu \right) r + \frac{\rho^{2}}{r} \right\rbrack - \rho^{2} \sigma_{\rho} \left\lbrack \left( 1 - 2\nu \right) r + \frac{\eta^{2}}{r^{2}} \right\rbrack \right\} - \nu z_{\text{o}}^{f} r

.. math::
    :label: 8.2-25

	u_{\text{th}}\left( r \right) &= \text{ Displacements due to thermal expansion} \\
    \ &= \left( \frac{1 + \nu}{1 - \nu} \right) \left\{ r I \left( r \right) + \frac{I \left( \eta \right)\eta^{2}}{\left( \eta^{2} - \rho^{2} \right)} \left\lbrack \left( 1 - 2\nu \right) r + \frac{\rho^{2}}{r} \right\rbrack \right\} - \nu z_{\text{o}}^{th} r

where :math:`z_{\text{o}}` has been divided into its thermal and boundary force
components, :math:`z_{\text{o}}^{th}` and
:math:`z_{\text{o}}^{f}` respectively (see :numref:`section-8.2.4`).

The stresses are similarly divided into their thermal and externally
applied force components.

.. math::
    :label: 8.2-26

	\sigma_{\text{r}} \left( r \right) = \sigma_{\text{r}}^{f} \left( r \right) + \sigma_{\text{r}}^{\text{th}} \left( r \right)

.. math::
    :label: 8.2-27

	\sigma_{\text{r}}^{f}\left( r \right) &= \text{ Radial stress from externally applied forces} \\
    \ &= \frac{1}{\left( \eta^{2} - \rho^{2} \right)} \left\lbrack \eta^{2} \sigma_{\eta}\left( 1 - \frac{\rho^{2}}{r^{2}} \right) - \rho^{2} \sigma_{\rho} \left( 1 - \frac{\eta^{2}}{r} \right) \right\rbrack

.. math::
    :label: 8.2-28

	\sigma_{\text{r}}^{\text{th}}\left( r \right) &= \text{ Radial stress from thermal expansion} \\
    \ &= \frac{E}{\left( 1 - \nu \right)} \left\lbrack \frac{\eta^{2} I \left( \eta \right)}{\left( \eta^{2} - \rho^{2} \right)}\left( 1 - \frac{\rho^{2}}{r^{2}} \right) - I \left( r \right) \right\rbrack

.. math::
    :label: 8.2-29

	\sigma_{\theta} \left( r \right) = \sigma_{\text{r}}^{f} \left( r \right) = \sigma_{\text{r}}^{th} \left( r \right)

.. math::
    :label: 8.2-30

	\sigma_{\theta}^{f}\left( r \right) &= \text{ Circumferential stress from externally applied forces} \\
    \ &= \frac{1}{\left( \eta^{2} - \rho^{2} \right)} \left\lbrack \eta^{2} \sigma_{\eta}\left( 1 + \frac{\rho^{2}}{r^{2}} \right) - \rho^{2} \sigma_{\rho} \left( 1 + \frac{\eta^{2}}{r^{2}} \right) \right\rbrack

.. math::
    :label: 8.2-31

	\sigma_{\text{r}}^{th}\left( r \right) &= \text{ Circumferential stress from thermal expansion} \\
    \ &= \frac{E}{\left( 1 - \nu \right)} \left\lbrack \frac{\eta^{2} I \left( \eta \right)}{\left( \eta^{2} - \rho^{2} \right)}\left( 1 + \frac{\rho^{2}}{r^{2}} \right) + I \left( r \right) - \Delta\left( \alpha T \right) \right\rbrack

Throughout the derivation given above, only the thermal components
contain the explicit reference to a change from one state to another,
i.e., reliance on temperatures :math:`T_{2}` and :math:`T_{1}` of
:eq:`8.2-5`. All the force components are based on a change from a zero stress
state to some state created by the imposition of the external force
boundary conditions. In the calculational procedure used in SAS4A, there
are a series of time steps with different conditions at the beginning
and end of the step. As mentioned above, the SAS4A thermal hydraulic
routines generate temperatures at the beginning and end of the time step
and then DEFORM‑4 determines the changes in dimensions, stress state,
and characterization that occur during the time step. In order to use
this natural, incremental approach, :eq:`8.2-24`, :eq:`8.2-27`, and :eq:`8.2-30` must
be changed to represent the changes from the beginning of the time step
to the end.

:eq:`8.2-27` gives the radial stress state set up by the externally
applied stresses :math:`\sigma_{\eta}` and :math:`\sigma_{\rho}` when the material was
initially in an unstressed condition. If a different set of conditions
existed, then a different stress state is set up. The change from one
state to the next can be found by using :eq:`8.2-27` twice, once with each
external stress boundary condition, and then subtracting.

.. math::
    :label: 8.2-32

	\Delta\sigma_{\text{r,i}}^{f} \left( r \right) = \frac{1}{\left( \eta^{2} - \rho^{2} \right)}\left\lbrack \eta^{2} \left( 1 - \frac{\rho^{2}}{r^{2}} \right) \left( \sigma_{\eta,\text{i}} - \sigma_{\eta,\text{i} - 1} \right) - \rho^{2} \left( 1 - \frac{\eta^{2}}{r^{2}} \right) \left( \sigma_{\rho,\text{i}} - \sigma_{\rho,\text{i} - 1} \right) \right\rbrack

where

:math:`\sigma_{\rho,\text{i}}`, :math:`\sigma_{\eta,1}` = Inner and outer externally
applied stresses at the end of time step :math:`i`, respectively

:math:`\sigma_{\rho,\text{i} - 1} \sigma_{\eta,\text{i} - 1}` = Inner and outer
externally applied stresses at the beginning of time step :math:`i` (end of time
step :math:`i-1`), respectively

:math:`\Delta\sigma_{\text{r,i}}^{f}(r)` = Incremental change in the radial
stress due to changes in the externally applied stresses :math:`\sigma_{\rho}` and
:math:`\sigma_{\eta}`

In a similar manner, :eq:`8.2-24` and :eq:`8.2-30` can be used to provide the
changes occurring during the computational time step.

.. math::
    :label: 8.2-33

    \Delta u_{\text{f,i}} \left( r \right) = \frac{\left( 1 + v \right)}{\left( \eta^{2} - \rho^{2} \right) E} \left\{ \eta^{2} \left( \sigma_{\text{n,j}} - \sigma_{\eta,i - 1} \right)\left\lbrack \left( 1 - 2\nu \right) r + \frac{\rho^{2}}{r} \right\rbrack \right. \\
    \left. - \rho^{2} \left( \sigma_{\rho,\text{i}} - \sigma_{\rho \text{i} - 1} \right)\left\lbrack \left( 1 - 2\nu \right) r + \frac{\eta^{2}}{r} \right\rbrack \right\} - \nu \Delta z_{\text{o,i}}^{f} r

.. math::
    :label: 8.2-34

    \Delta \sigma_{\theta,\text{i}}^{f} \left( r \right) = \frac{\left( 1 \right)}{\left( \eta^{2} - \rho^{2} \right)} \left\lbrack \eta^{2} \left( 1 + \frac{\rho^{2}}{r^{2}} \right)\left( \sigma_{\eta,\text{i}} - \sigma_{\eta,\text{i} - 1} \right) \right. \\
    \left. - \rho^{2} \left( 1 + \frac{\eta^{2}}{r^{2}} \right)\left( \sigma_{\rho,\text{i}} - \sigma_{\rho,\text{i} - 1} \right) \right\rbrack

where

:math:`\Delta u_{\text{f,i}}\left( r \right)` = Incremental elastic displacement due to
changes in the externally applied stresses :math:`\sigma_{\rho}` and
:math:`\sigma_{\eta}`

:math:`\Delta{\sigma_{\theta,\text{i}}^{f}} \left( r \right)` = Incremental change in
the circumferential stress due to changes in the externally applied
stresses :math:`\sigma_{\rho}` and :math:`\sigma_{\text{v}}`

:math:`\Delta{z_{\text{o,i}}^{f}}` = Incremental change in the axial plane
strain due to changes in the external applied stresses :math:`\sigma_{\rho}` and
:math:`\sigma_{\eta}`

As mentioned above, this same procedure is not required for the
thermally induced stresses and strains, since they are derived from
explicit temperature changes. With the initial and final temperatures in
:eq:`8.2-5` defined as the temperatures at the beginning and end of the
current time step, respectively, the incremental changes in the stresses
and strains are determined.

:eq:`8.2-25`, :eq:`8.2-28`, and :eq:`8.2-31`, define the incremental changes in
the deformation, radial stress, and circumferential stress in response
to a temperature change during the time step. :eq:`8.2-32` through
:eq:`8.2-34` define the changes in response to changes in the externally
applied forces. The incremental changes in the axial stresses can be
found from :eq:`8.2-4`, once the axial plane strain is determined. The
separation of the stresses into the thermal and force components makes
it possible to implement thermal stress relaxation in a straightforward
manner.

The subroutine FSIGMA solves the equations for the fuel and the
subroutine CSIGMA solves them for the cladding.

.. _section-8.2.2:

Cracked Fuel Thermoelastic Solution and Crack Volume
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If the circumferential stress at a cell boundary exceeds the fracture
strength of the fuel, the cell immediately inside that boundary is
assumed to crack. The new outer solid fuel boundary is then studied to
determine if it will also crack. This process is repeated until a stable
solid boundary is reached, or cracking occurs to the central void or
molten fuel boundary. This procedure is carried out in the subroutine
FSIGMA. The treatment for the solid zone was presented previously and
the solution in the cracked zone is given below.

In the cracked zone, it is assumed that numerous radial and transverse
cracks exist and extend inward to the same radial position. Under these
conditions the circumferential and axial stress, :math:`\sigma_{\theta}` and
:math:`\sigma_{\text{z}}`, respectively, are both set equal to the negative of the
plenum gas pressure, :math:`P_{\text{g}}`, since it is assumed that communication
exists with the plenum.

.. math::
    :label: 8.2-35

	\sigma_{\theta} = \sigma_{\text{z}} = - P_{\text{g}}

If these values are substituted into the equilibrium :eq:`8.2-11`, and the
integration performed from the outer fuel radius, :math:`R_{\text{f}}`, to a
radius :math:`r`, the radial stress at any point in the cracked region
results:

.. math::
    :label: 8.2-36

	\sigma_{\text{r}}\left( r \right) = - P_{\text{g}} + \frac{R_{\text{f}}}{r} \left( P_{\text{g}} + \sigma_{\text{FC}} \right)

where

:math:`\sigma_{\text{FC}}` = Fuel-cladding interface stress, the
negative of the interface pressure

If the integration is performed outward from the boundary between the
continuous and the cracked fuel, :math:`\eta`, to a radius :math:`r`, then the stress in
the cracked fuel is of the form:

.. math::
    :label: 8.2-37

	\sigma_{\text{r}}\left( r \right) = - P_{\text{g}} + \frac{\eta}{r} \left( P_{\text{g}} + \sigma_{\eta} \right)

where

:math:`\sigma_{\eta}` = Stress at the outer boundary of the solid fuel

Since :eq:`8.2-36` and :eq:`8.2-37` must define the same stress at a given
radius, the relationship between the stress at the solid‑cracked
boundary and the fuel‑cladding interface may be determined as

.. math::
    :label: 8.2-38

	\sigma_{\eta} = - P_{\text{g}} + \frac{R_{\text{f}}}{\eta} \left( P_{\text{g}} + \sigma_{\text{FC}} \right)

It should be noted that the radial stress in the cracked region contains
no dependence on the thermal expansion of the region.

To obtain the radial displacement, :math:`u_{\text{r}}`, :eq:`8.2-35` and :eq:`8.2-36`
are substituted into :eq:`8.2-2` and the result used in :eq:`8.2-6`. Upon
integration, the radial displacement is given by

.. math::
    :label: 8.2-39

	u_{\text{r}} = u_{\eta} + \left( 2\nu - 1 \right) \left( r - \eta \right) \frac{P_{\text{g}}}{E_{\text{c}}} + \frac{\left( P_{\text{g}} - \sigma_{\eta} \right)}{E_{\text{c}}} \eta \ln \left( \frac{r}{\eta} \right) + \int_{\eta}^{r}{\Delta \left( \alpha T \right) \text{dr}'}

where

:math:`u_{\text{r}}` = Displacement at radius :math:`r` in the cracked fuel zone

:math:`u_{\eta}` = Displacement of the outer surface of the continuous
fuel zone

:math:`E_{\text{c}}` = Mass averaged modulus of elasticity in the cracked fuel
zone

As in the section on solid fuel or cladding, the above derivations
contain the implicit assumption that the cells start out in an
unexpanded, stress free state. This is not what exists at the beginning
of the time step. :eq:`8.2-35` through :eq:`8.2-39` can be used to obtain
the changes that occur from the beginning of the time step to the end.

.. math::
    :label: 8.2-40

	\Delta\sigma_{\theta,\text{i}} = \Delta\sigma_{\text{z,i}} =  - \Delta P_{\text{g,i}}

.. math::
    :label: 8.2-41

	\Delta\sigma_{\text{r,i}} = - \Delta P_{\text{g,i}} + \frac{R_{\text{f}}}{r}\left( \Delta P_{\text{g,i}} + \Delta\sigma_{\text{FC,i}} \right) \\
    = - \Delta P_{\text{g,i}} + \frac{\eta}{r} \left( \Delta P_{\text{g,i}} + \Delta\sigma_{\text{n,i}} \right)

.. math::
    :label: 8.2-42

	\Delta\sigma_{\text{n,i}} = - \Delta P_{\text{g,i}} + \frac{R_{\text{f}}}{r}\left( \Delta P_{\text{g,i}} + \Delta\sigma_{\text{FC,i}} \right)

where

:math:`\Delta P_{\text{g,i}}` = Change in plenum pressure during time step :math:`i`

:math:`\Delta\sigma_{\text{FC,i}}` = Change in fuel-cladding interface stress
during time step :math:`i`

:math:`\Delta\sigma_{\text{n,i}}` = Change in stress at solid fuel-cracked fuel
boundary during time step :math:`i`

The displacement calculated in :eq:`8.2-39` contains two parts, that due
to forces and that due to temperature changes. As with the solid fuel,
the thermal effects are already handled in an incremental fashion, so no
changes are required. However, the force effects have to be modified to
handle the changes during the time step. Using the solid fuel results
for incremental changes, the value of outer solid fuel displacement,
:math:`u_{\eta}`, will be an incremental change. The force effects can be
made incremental in the manner used above.

.. math::
    :label: 8.2-43

	u_{\text{r,i}} = u_{\eta,\text{i}} \left( 2\nu - 1 \right) \left( r - \eta \right) \frac{\Delta P_{\text{g,i}}}{E_{\text{c}}} + \frac{\left( \Delta P_{\text{g,i}} + \Delta\sigma_{\text{n,i}} \right)}{E_{\text{c}}} \eta \ln \left( \frac{r}{\eta} \right) + \int_{\eta}^{r}{\Delta \left( \alpha T \right) \text{dr}'}

:eq:`8.2-43` gives the displacements of the cracked fuel nodes in the
incremental manner desired. These equations are solved in the subroutine
FSIGMA.

As the dimensions of a cracked fuel cell change, so will the fraction of
the volume that is associated with the cracks. These changes are
calculated in the subroutine CRAKER. In the current version of DEFORM‑4,
the volume associated with transverse, or axial, cracking is neglected.
The radial crack volume fraction is affected by three factors: (1)
changes in the cell boundary locations, (2) circumferential strain, and
(3) fission‑product‑induced fuel swelling. The first two processes are
treated in the subroutine CRAKER, whereas the third is treated in the
subroutine FSWELL.

.. math::
    :label: 8.2-44

	\Delta V_{\text{CRK}} = \Delta V \left( u \right) \epsilon_{\theta}^{c} \Delta V_{\text{s}}

where

:math:`\Delta V_{\text{CRK}}` = Fractional volume change due to radial
cracks

:math:`\Delta V \left( u \right)` = Fractional volume change associated
with the changes in the radial boundaries of the cell by the constant
displacement :math:`u`

:math:`\epsilon_{\theta}^{c}` = Circumferential strain of the cracked
fuel cell

:math:`\Delta V_{\text{s}}` = Fractional volume change associated with
fission‑product swelling in the cell

The first term in :eq:`8.2-44` is based on purely geometric
considerations. If an annulus with inner radius :math:`r_{\text{i}}` and outer
radius :math:`r_{\text{o}}` is moved radially by an amount :math:`u`, the new inner
and outer radius become :math:`r_{\text{i}} + u` and :math:`r_{\text{o}} + u`,
respectively. This new annulus has a different volume from the original
configuration. The change in volume fraction is assumed to be contained
in the cracks in the fuel cell.

.. math::
    :label: 8.2-45

	\Delta V \left( u \right) = \frac{\left\lbrack \left( r_{\text{o}} + u \right)^{2} - \left( r_{\text{i}} + u \right)^{2} \right\rbrack - \left\lbrack r_{\text{o}}^{2} + r_{\text{i}}^{2} \right\rbrack}{r_{0}^{2} - r_{\text{i}}^{2}}

:eq:`8.2-45` reduces to

.. math::
    :label: 8.2-46

	\Delta V \left( u \right) = \frac{2u}{r_{\text{o}} - r_{\text{i}}}

In the determination of the circumferential strain in a cracked fuel
cell, it is assumed that the fuel continues to act as linear elastic
material obeying :eq:`8.2-3`. With the circumferential and axial stresses
set to the negative of the plenum pressure, :eq:`8.2-35`, and the radial
stress as given in :eq:`8.2-36`, the circumferential strain in the cracked
zone can be determined as the sum of the force and thermal component to
the strain :math:`\varepsilon_{\theta,\text{f}}^{c}` and
:math:`\varepsilon_{\theta,\text{th}}^{c}`, respectively.

.. math::
    :label: 8.2-47

	\varepsilon_{\theta}^{c} = \varepsilon_{\theta,\text{f}}^{c} + \varepsilon_{\theta,\text{th}}^{c}

.. math::
    :label: 8.2-48

	\varepsilon_{\theta,\text{f}}^{c} = - \left( 1 - 2\nu \right) \frac{\Delta P_{\text{g,i}}}{E_{\text{c}}} - \frac{\nu R_{\text{f}}}{E_{\text{c}}r} \left( \Delta P_{\text{g,i}} + \Delta\sigma_{\text{FC,i}} \right)

.. math::
    :label: 8.2-49

	\varepsilon_{\theta,\text{th}}^{c} = \Delta \left( \alpha T \right)

The value of the force strain used in determining the strain of a
particular cell is the average of the strain at the cell boundaries and
is based on the changes occurring during the time step. The thermal
component is evaluated from the change in the cell temperature during
the time step.

.. _section-8.2.3:

Fully Cracked Fuel
~~~~~~~~~~~~~~~~~~

In some pretransient situations and many transient cases, it is possible
for the solid fuel to become fully cracked, i.e., there is no solid fuel
annulus. Being fully cracked, there would be no resistance to radial
relocation outward, until the cladding is reached, or radially inward,
until the crack volume is closed in the central solid regions of the
fuel. If the pressure in the cracks remains the plenum pressure,
:eq:`8.2-38` then represents the necessary equilibrium condition that must be
satisfied. However, if the crack pressure is assumed to reach a level
equivalent to the central cavity pressure, the substitution in
:eq:`8.2-11` yields a constant pressure, equivalent to the cavity pressure,
throughout the fuel and as the fuel-cladding interface pressure. Both
options are available within DEFORM‑4.

The movement of the cracked fuel is controlled by the two external
forces applied: (1) the central cavity pressure, and (2) the
fuel‑cladding interface pressure. If no molten cavity exists with a
pressure greater than the plenum pressure, the cracked fuel cannot
relocate radially outward to remove the fuel‑cladding gap that may
exist. Under these conditions the maximum fuel-cladding interface
pressure is equivalent to the plenum pressure. If the previous time step
contained a solid fuel annulus which had produced an interface pressure
greater than the plenum pressure, once full cracking is achieved the
interface pressure will drop to the plenum pressure, and the fuel may be
relocated inward in response to the alleviation of previous elastic
strains in the cladding.

If a pressurized molten cavity does exist at the time the cracked region
reaches the melt boundary, or melting proceeds to the cracked boundary,
then radial relocation to the cladding surface will take place if a gap
existed. If no gap existed, then the relocation could occur either
outward or inward depending on the previous interface pressure, the
cavity pressure, and the option used for the pressure of the gas in the
cracks.

When the crack pressure is assumed to remain at the plenum pressure, the
new interface pressure is determined from :eq:`8.2-38`.

.. math::
    :label: 8.2-50

	\sigma_{\text{FC}} = - P_{\text{g}} + \frac{R_{\text{cav}}}{R_{\text{f}}} \left( \sigma_{\text{cav}} + P_{\text{g}} \right)

where

:math:`\sigma_{\text{cav}}` = Molten central cavity pressure

:math:`R_{\text{cav}}` = Outer radius of the molten central cavity

If this new interface pressure is less than the previous time step,
removal of the previous elastic strain will take place, moving the
cracked fuel toward the center. If it is greater than the previous
interface pressure, more elastic strain will be produced, and the fuel
and cladding will move outward. If this new pressure produces a
circumferential stress larger than the cladding flow stress, the
cladding will strain plastically to provide enough volume to reduce the
cavity pressure to a value that produces an interface pressure equal to
that necessary to produce a circumferential stress equal to the flow
stress.

These conditions are all handled by the fully crack fuel solution driver
subroutine MKDRIV.

With these considerations it is possible for the cladding stress to
build up while solid fuel exists, then become alleviated when the fuel
becomes fully cracked with a low molten cavity pressure, and then again
build up as the molten cavity pressure and melt radius increase. The
timing of such behavior and the magnitude reached will depend on the
particular transient being studied.

.. _section-8.2.4:

Axial Expansion Solution
~~~~~~~~~~~~~~~~~~~~~~~~

As discussed in :numref:`section-8.2.1`, the analytical solution to the mechanics
equations is produced through a generalized plane strain assumption. The
axial interfaces between segments are assumed to remain parallel, and a
segment expands or contracts with a uniform strain, :math:`z_{\text{o}}`, over
its entire radius. Since this axial strain exists in the equations which
represent the radial displacement function, it is necessary to find the
axial strain prior to the radial strain results. The calculation for the
axial strain of a segment is performed in the subroutine EXPAND.

The axial force in the fuel segment, :math:`F_{\text{f}}`, is found by
integrating the axial stress, :math:`\sigma_{\text{z}}`, over the cylindrical fuel
geometry,

.. math::
    :label: 8.2-51

	F_{\text{f}} = 2 \pi \int_{\rho}^{\eta}\sigma_{\text{z}} r \text{dr}

Substituting :eq:`8.2-27`, :eq:`8.2-28`, :eq:`8.2-30`, :eq:`8.2-31` and :eq:`8.2-10` into :eq:`8.2-4` and
solving for the axial stress function, yields:

.. math::
    :label: 8.2-52

	\sigma_{\text{z}} = \frac{2\nu E}{\left( \eta^{2} - \rho^{2} \right)} \left\lbrack \frac{I\left( \eta \right) \eta^{2}}{\left( 1 - \nu \right)} - \frac{\left( \rho^{2} - \sigma_{\rho} - \eta^{2} \sigma_{\eta} \right)}{E} \right\rbrack - \frac{E\Delta\left( \alpha T \right)}{\left( 1 - \nu \right)} + E z_{O}

This is then used in :eq:`8.2-51` and integrated to yield

.. math::
    :label: 8.2-53

	F_{\text{f}} = - 2 \pi E \eta^{2} I \left( n \right) + 2 \pi \nu \left( \eta^{2} \sigma_{\eta} - \rho^{2} \sigma_{\rho} \right) + \pi E z_{O} \left( \eta^{2} - \rho^{2} \right)

In order to find the axial plane strain, a total force balance is
performed. The state of the fuel‑cladding gap can influence the terms in
the force summation. If the fuel‑cladding gap is open, or the free axial
expansion option is chosen through the input parameter NAXOP, then the
force summation contains no term for the cladding effects. If, however,
the fuel and cladding are in contact, then cladding terms must be
included in the force balance. The following is the general equation for
the force balance.

.. math::
    :label: 8.2-54

	F_{\text{f}} = F_{\text{cav}} + F_{\text{ax}} + F_{\text{c}}

where

:math:`F_{\text{cav}}` = Force in the central void or molten fuel cavity

:math:`F_{\text{ax}}` = Force applied axially to the fuel column,
usually from the plenum gas pressure

:math:`F_{\text{f}}` = Force from the solid fuel zone

:math:`F_{\text{c}}` = Force from the cladding, which is 0 with free axial
expansion

In the central void or molten cavity, the force is given by

.. math::
    :label: 8.2-55

	F_{\text{cav}} = \pi \rho^{2} P_{\text{cav}}

where

:math:`P_{\text{cav}}` = The pressure in the molten cavity or central void

:math:`\rho` = The radial extent of melting or the central void radius

The axial force from the plenum pressure is given by

.. math::
    :label: 8.2-56

	F_{\text{ax}} = \pi r_{\rho}^{2} P_{\text{gas}}

where

:math:`P_{\text{gas}}` = Fission-gas plenum pressure

:math:`r_{\text{p}}` = Radius of the plenum

For the cladding, the force is either zero or the same as :eq:`8.2-53` but
with cladding properties, thermal expansion, and inner and outer
boundary forces.

If free axial expansion is assumed, then :eq:`8.2-53`, :eq:`8.2-55`, and :eq:`8.2-56` can
be used in :eq:`8.2-54`. This is then solved for the uniform axial strain.

.. math::
    :label: 8.2-57

	z_{\text{o}} = \left\lbrack z_{\text{o}}\left( \text{thermal} \right) + z_{\text{o}} \left( \text{ forces } \right) \right\rbrack A_{\text{f}}

.. math::
    :label: 8.2-58

	z_{\text{o}}\left( \text{thermal} \right) = \frac{2 {\eta}^{2} I \left( \eta \right)}{\left( {\eta}^{2} - {\rho}^{2} \right)}

.. math::
    :label: 8.2-59

	z_{\text{o}}\left( \text{forces} \right) = \frac{- 2 v}{\left( {\eta}^{2} - \rho^{2} \right)} \frac{\left( {\eta}^{2} \sigma_{\eta} - \rho^{2} \sigma_{\rho} \right)}{E} - \frac{\left( \rho^{2} P_{\text{cav}} + r_{\rho}^{2} P_{\text{gas}} \right)}{E}

where

:math:`A_{\text{f}}` = Fraction of axial expansion to be used

All properties in the above two equations refer to the fuel, and the
modulus of elasticity, :math:`E`, is a mass-averaged value.

Since this equation assumes an initial stress free state, it must be
modified to account for the changes, which take place during the
specific time step. The time step changes occur in the inner and outer
boundary conditions, :math:`\sigma_{\rho}` and :math:`\rho_{\eta}`, and the cavity and
plenum pressures, :math:`P_{\text{CAV}}` and :math:`P_{\text{gas}}`.

.. math::
    :label: 8.2-60

	z_{\text{o}}\left( \text{forces} \right) = \frac{- 2 v}{\eta^{2} - \rho^{2}}\frac{\left( \eta^{2}\Delta\sigma_{\eta,\text{i}} - \rho^{2}\Delta\sigma_{\rho,\text{i}} \right)}{E} - \frac{\left( \rho^{2} P_{\text{cav}} + r_{\rho}^{2} P_{\text{gas}} \right)}{E \left( \eta^{2} - \rho^{2} \right)}

:eq:`8.2-60` together with :eq:`8.2-58` define the axial strain
occurring during the time step due to force and thermal considerations,
respectively.

For the case where the fuel and cladding are considered "constrained",
the fuel and cladding surfaces are assumed to be locked to each other.
:eq:`8.2-54` is used with :eq:`8.2-53` (used twice, once with fuel
properties, once with cladding properties), and :eq:`8.2-55` and :eq:`8.2-56`. The
result is again solved for the axial strain

.. math::
    :label: 8.2-61

	z_{\text{o}}\left( \text{thermal} \right) = \frac{2 \left\lbrack E_{\text{f}} \eta_{\text{f}}^{2} I \left( n_{\text{f}} \right) + E_{\text{c}} \eta_{\text{c}}^{2} I \left( \eta_{\text{c}} \right) \right\rbrack}{\left\lbrack E_{\text{c}}\left( \eta_{\text{c}}^{2} - \rho_{\text{c}}^{2} \right) + E_{\text{f}}\left( \eta_{\text{f}}^{2} - \rho_{\text{f}}^{2} \right) \right\rbrack}

.. math::
    :label: 8.2-62

	z_{0}\left( \text{forces} \right) = \frac{- 2 \left\lbrack v_{\text{f}}\left( \eta_{\text{f}}^{2}\sigma_{\eta,\text{f}} - \rho_{\text{f}}^{2}\sigma_{\rho,\text{f}} \right) + v_{\text{c}}\left( \eta_{\text{c}}^{2}\sigma_{\eta,\text{c}} - \rho_{\text{c}}^{2}\sigma_{\rho,\text{c}} \right) \right\rbrack - \left( \rho_{\text{f}}^{2} P_{\text{cav}} + r_{\text{p}}^{2} P_{\text{gas}} \right)}{E_{\text{c}}\left( \eta_{\text{c}}^{2} - \rho_{\text{c}}^{2} \right) + E_{\text{f}}\left( \eta_{\text{f}}^{2} - \rho_{\text{f}}^{2} \right)}

The subscript "f" refers to fuel properties and dimensions and the
subscript "c" refers to the cladding properties and dimensions. Again,
:eq:`8.2-62` must be transformed to consider the changes from one time
step to the next.

.. math::
    :label: 8.2-63

	z_{0}\left( \text{forces} \right) = \frac{- 2 \left\lbrack v_{\text{f}}\left( \eta_{\text{f}}^{2}\Delta \sigma_{\eta,\text{f}} - \rho_{\text{f}}^{2} \Delta \sigma_{\rho,\text{f}} \right) + v_{\text{c}}\left( \eta_{\text{c}}^{2}\Delta \sigma_{\eta,\text{c}} - \rho_{\text{c}}^{2}\Delta\sigma_{\rho,\text{c}} \right) \right\rbrack - \left( \rho_{\text{f}}^{2} \Delta P_{\text{cav}}
	+ r_{\text{p}}^{2}\Delta P_{\text{gas}} \right)}{E_{\text{c}}\left( \eta_{\text{c}}^{2} - \rho_{\text{c}}^{2} \right) + E_{\text{f}}\left( \eta_{\text{f}}^{2} - \rho_{\text{f}}^{2} \right)}

:eq:`8.2-63` together with :eq:`8.2-61` defines the axial strain for
the time step when the fuel and cladding are locked together.

The axial strain is separated into its thermal and force components to
allow for the option to use only thermal effects or both thermal and
force effects through the input parameter IAXTHF. There are also four
options available which concern the treatment of the fuel‑cladding
locking. The axial expansion can be set to the free axial expansion of
the fuel or the cladding, to always be constrained expansion, or to be a
combination depending on the actual interface conditions. In the fourth
option, all axial segments below the top locked segment are assumed to
be in the constrained state. These options are controlled through the
input parameter NAXOP. The most realistic options are both thermal and
force components with the mixed free/constrained state. The other
options are available to facilitate the study of axial expansion
assumptions on accident sequences.

There also may be a third fuel‑cladding boundary condition, which does
not currently exist in DEFORM‑4. This is the "slip" condition where, at
some level of mismatch between the fuel and cladding forces, the fuel
would slide along the cladding. This is noted for possible inclusion in
the future.

In the equations given above, the implicit assumption is present that
the fuel in the region under consideration is both solid and uncracked.
In the current version of DEFORM‑4, the entire solid fuel zone is used
in the calculation, both continuous and cracked. The result is to
provide average axial expansion effects in a generalized plane strain
framework. If transverse crack volumes are instituted in the cracking
model, then the expansion effects in the cracked fuel region will be
restudied. In order to provide a conservative assumption, when the
melting reaches the cracked fuel boundary, axial expansion of that node
is stopped. This is used to handle the necessity of expanding into
transverse crack volume, before additional reactivity effects are
treated.