Presentation of the model ====================== Equilibrium proportion ------------------------ Zircaloy has a compact hexagonal structure called phase :math:`\mathrm{\alpha }`, which is stable up to temperatures of the order of :math:`700°C`. Beyond about :math:`700°C`, an allotropic transformation begins to a :math:`\beta` cubic phase, which is complete around :math:`975°C`. The proportion of phase :math:`{Z}_{\mathrm{\beta }}^{\mathit{eq}}` at equilibrium is given by the following equation, of the Johnson-Mehl-Avrami type: .. math:: :label: eq-1 \ {\ begin {array} {ccc} {Z} _ {\ mathrm {\ beta}}} ^ {\ mathit {eq}} =0&\ text {for} & T\ le {T} _ {d} _ {d}} ^ {\ mathit {eq} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {thrm {exp}\ left (- {(K (T- {T} _ {d} _ {d} ^ {\ mathit {eq}}))} ^ {n}\ right) &\ text {for} & {T} _ {d} _ {d}} ^ {d} ^ {\ mathit {eq}} ^ {\ mathit {eq}}}\\ {Z}} _ {\ mathrm {\ beta}} ^ {\ mathit {eq}} =1&\ text {for} & T> {T} _ {f} ^ {\ mathit {eq}}}\ end {array}}\ end {array} where :math:`{T}_{d}^{\mathit{eq}}` is the starting temperature :math:`\mathrm{\alpha }\iff \mathrm{\beta }` at equilibrium, :math:`{T}_{f}^{\mathit{eq}}` the transformation end temperature at equilibrium, the temperature at the end of transformation at equilibrium, :math:`T` the temperature and :math:`K` and :math:`n` two material parameters. Equivalently, by inverting the equation, we obtain the equivalent temperature :math:`{T}^{\mathit{eq}}` as a function of the :math:`{Z}_{\mathrm{\beta }}` proportion of phase :math:`\mathrm{\beta }`: .. math:: :label: eq-2 \ {\ begin {array} {cc} {T} {T} ^ {\ mathit {eq}} ^ {T} _ {d} ^ {\ mathit {eq}} +\ frac {1} {K} {K} {\ left (\ mathrm {log}} ^ {\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left (\ frac {1}} {1} {Z} _ {\ mathrm {\ beta}}}}\ right)\ right)} ^ {1/n} &\ text {for} 0\ le {Z} _ {\ mathrm {\ beta}}\ le\ mathrm {0.99}\\ {T} ^ {\ mathit {eq}}} = {T} {eq}} = {T} _ {f}} _ {f} ^ {\ f} ^ {\ mathrm {eq}} ^ {\ mathrm {eq}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}}} = {T}}} = {T}}} = {T}}} = {thrm {\ beta}}\ le 1\ end {array} The temperature at the end of transformation at equilibrium :math:`{T}_{f}^{\mathit{eq}}` is chosen such that corresponding to a proportion :math:`\mathrm{0,99}` of phase :math:`\mathrm{\beta }` transformed, i.e.: .. math:: :label: eq-3 {T} _ {f} ^ {\ mathit {eq}}} = {T} _ {eq}} = {T} _ {d} ^ {\ mathit {eq}} +\ frac {1} {K} {\ left (\ mathrm {log}}\ left}\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}} Evolution equation under heating --------------------------------- The transformation when heated is transformation :math:`\mathrm{\alpha }\Rightarrow \mathrm{\beta }`. The temperature at which phases are transformed upon heating :math:`{T}_{c}` depends on the temperature rate upon heating and is given by the equation: .. math:: : label: eq-4 {T} _ {c} = {T} _ {c} {c} ^ {1} {({V} _ {\ mathit {ch}})}} ^ {{T} _ {c} {c} ^ {2}} With :math:`{V}_{\mathit{ch}}` the heating rate in :math:`°C/s` and :math:`{T}_{c}\ge {T}_{d}^{\mathit{eq}}`. The evolution model of phase :math:`\mathrm{\beta }` during heating is given by the following differential equation (Holt model): .. csv-table:: ":math:`\frac{{\mathit{dZ}}_{\mathrm{\beta }}}{\mathit{dt}}={A}_{c}\mathrm{exp}\left(-\frac{E}{RT}\right){", "T-{T}^{\mathit{eq}}({Z}_{\mathrm{\beta }})", "}^{M}` ", "(5)" :math:`{T}^{\mathit{eq}}({Z}_{\mathrm{\beta }})` is the equilibrium temperature corresponding to the instantaneous proportion :math:`{Z}_{\mathrm{\beta }}` of phase :math:`\mathrm{\beta }` and given by the equation. :math:`{T}_{c}^{1}`, :math:`{T}_{c}^{2}`, :math:`{A}_{c}`, :math:`\frac{E}{R}`, and :math:`M` are material parameters. Cooling evolution equation ------------------------ The transformation to cooling is transformation :math:`\mathrm{\beta }\Rightarrow \mathrm{\alpha }`. The temperature at the beginning of phase transformation upon cooling :math:`{T}_{r}` depends on the temperature rate upon cooling and is given by the equation: .. math:: :label: eq-6 {T} _ {r} = {T} _ {r} ^ {r} ^ {1} + {T} _ {r} ^ {2}\ mathrm {ln} ({V} _ {\ mathit {ref}}) With :math:`{V}_{\mathit{ref}}` the cooling rate in :math:`°C/s` and :math:`{T}_{r}\le {T}_{f}^{\mathit{eq}}`. The evolution model from phase :math:`\mathrm{\beta }` to cooling is given by the following differential equation: .. csv-table:: ":math:`\frac{{\mathit{dZ}}_{\mathrm{\beta }}}{\mathit{dt}}=-", "T-{T}^{\mathit{eq}}", "\mathrm{exp}\left({A}_{r}+{B}_{r}", "T-{T}^{\mathit{eq}}", "\right){Z}_{\mathrm{\beta }}(1-{Z}_{\mathrm{\beta }})` for :math:`{T}_{d}^{\mathit{eq}}\le T\le {T}_{f}^{\mathit{eq}}` ", "(7)" :math:`{T}_{r}^{1}`, :math:`{T}_{r}^{2}`, :math:`{A}_{r}`, and :math:`{B}_{r}` are material parameters. Conditions of use of the metallurgical model for any temperature transients ------------------------------------------------------------------------------------------------- A few rules ~~~~~~~~~~~~~~~~ * During the calculations, if the proportion of phase :math:`\mathrm{\beta }` is strictly greater than :math:`\mathrm{0,99}`, we round to one. For a heating rate of less than :math:`0.1°C/s`, :math:`{T}_{c}={T}_{d}^{\mathit{eq}}` is used. * If :math:`0\le {Z}_{\mathrm{\beta }}\le \mathrm{0,99}`, the following rule should be applied: * * If :math:`T>{T}^{\mathit{eq}}\iff {Z}_{\mathrm{\beta }}<{Z}_{\mathrm{\beta }}^{\mathit{eq}}`, apply the model to heating (even if the temperature speed is negative) * If :math:`T<{T}^{\mathit{eq}}\iff {Z}_{\mathrm{\beta }}>{Z}_{\mathrm{\beta }}^{\mathit{eq}}`, apply the model to cooling (even if the temperature rate is positive) Algorithm ~~~~~~~~~~ We consider any transient of temperature :math:`T(t)`. .. csv-table:: "*Note: to calculate the transformation start temperatures at heating* :math:`{T}_{c}` *and at cooling* :math:`{T}_{r}` *, it is necessary to calculate the heating and cooling rates, respectively. To calculate them, we use the sliding secant method (and not the instantaneous speed), hence steps 1 and 2 below.*" .. csv-table:: "*Note: once the threshold temperatures* :math:`{T}_{c}` *or* :math:`{T}_{r}` *have been exceeded and as long as the transformation is not complete (heating or cooling), the evolution equations are integrated even if the temperature exceeds the threshold again.*" * *Step 1*: Search for the instant :math:`{t}_{d}^{\mathit{eq}}` (or :math:`{t}_{f}^{\mathit{eq}}`) corresponding to the start temperature :math:`{T}_{d}^{\mathit{eq}}` (or end :math:`{T}_{f}^{\mathit{eq}}`, respectively) of transformation at equilibrium. * Case where :math:`{Z}_{\mathrm{\beta }}=0` initially: looking for :math:`{t}_{d}^{\mathit{eq}}` * Case where :math:`{Z}_{\mathrm{\beta }}=1` initially: looking for :math:`{t}_{f}^{\mathit{eq}}` * *Step 2*: Search for the instant :math:`{t}_{c}` (or :math:`{t}_{r}`) corresponding to the transformation start temperature :math:`{T}_{c}` (or :math:`{T}_{r}`, respectively) using the sliding secant method: * Case where :math:`{Z}_{\mathrm{\beta }}=0` initially: find the moment when the temperature :math:`T(t)` exceeds :math:`{T}_{c}`. * If :math:`T(t)>{T}_{d}^{\mathit{eq}}`, we increment the time, we calculate :math:`{T}_{c}` and we test the following condition: .. math:: :label: eq-8 \ text {(C1)}\ mathrm {:} T (t) > {T} _ {c} > {T} _ {c} ^ {1} {\ left (\ frac {T (t) - {T} - {T} _ {d} _ {d} _ {eq}}} {\ mathit {eq}}}} {d} ^ {\ mathit {eq}}}\ right)} ^ {{T} _ {c} ^ {2}} If condition :math:`(C1)` is true we go to step (3). * If :math:`T(t)\le {T}_{d}^{\mathit{eq}}` without reaching :math:`{T}_{c}`, you must then update :math:`{t}_{d}^{\mathit{eq}}` by restarting step 1 from the current moment. * Case where :math:`{Z}_{\mathrm{\beta }}=1` initially: find the moment when the temperature :math:`T(t)` returns to :math:`{T}_{r}`. * If :math:`T(t)<{T}_{f}^{\mathit{eq}}`, we increment the time, we calculate :math:`{T}_{r}` and we test the following condition: .. csv-table:: ":math:`\text{(C2)}\mathrm{:}T(t)<{T}_{r}={T}_{r}^{1}+{T}_{r}^{2}\mathrm{ln}\left(\frac{", "T(t)-{T}_{f}^{\mathit{eq}}", "}{t-{t}_{f}^{\mathit{eq}}}\right)` ", "(9)" If condition :math:`(C2)` is true we go to step (3). * If :math:`T(t)\ge {T}_{f}^{\mathit{eq}}` without reaching :math:`{T}_{r}`, you must then update :math:`{t}_{f}^{\mathit{eq}}` by restarting step 1 from the current moment. * *Step 3*: Once :math:`{T}_{c}` (or :math:`{T}_{r}`) is reached, the evolution of the phase fraction :math:`\mathrm{\beta }` appearing is calculated step by step using the Holt equation (on heating) or the cooling equation following the sign of :math:`(Z-{Z}_{\mathit{eq}})` and this as long as the phase fraction :math:`\mathrm{\beta }` remains less than :math:`\mathrm{0,99}` and greater than zero. and even if one goes through a temperature peak. * *Step 4*: If during step 3, the phase fraction :math:`\beta` becomes equal to 1 (or 0), step 1 is repeated from the current instant.