2. Presentation of the model#

2.1. Equilibrium proportion#

Zircaloy has a compact hexagonal structure called phase \(\mathrm{\alpha }\), which is stable up to temperatures of the order of \(700°C\). Beyond about \(700°C\), an allotropic transformation begins to a \(\beta\) cubic phase, which is complete around \(975°C\).

The proportion of phase \({Z}_{\mathrm{\beta }}^{\mathit{eq}}\) at equilibrium is given by the following equation, of the Johnson-Mehl-Avrami type:

(2.1)#\[\begin{split} \ {\ begin {array} {ccc} {Z} _ {\ mathrm {\ beta}}} ^ {\ mathit {eq}} =0&\ text {for} & T\ le {T} _ {d} _ {d}} ^ {\ mathit {eq} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {\ mathit {eq}} ^ {thrm {exp}\ left (- {(K (T- {T} _ {d} _ {d} ^ {\ mathit {eq}}))} ^ {n}\ right) &\ text {for} & {T} _ {d} _ {d}} ^ {d} ^ {\ mathit {eq}} ^ {\ mathit {eq}}}\\ {Z}} _ {\ mathrm {\ beta}} ^ {\ mathit {eq}} =1&\ text {for} & T> {T} _ {f} ^ {\ mathit {eq}}}\ end {array}}\ end {array}\end{split}\]

where \({T}_{d}^{\mathit{eq}}\) is the starting temperature \(\mathrm{\alpha }\iff \mathrm{\beta }\) at equilibrium, \({T}_{f}^{\mathit{eq}}\) the transformation end temperature at equilibrium, the temperature at the end of transformation at equilibrium, \(T\) the temperature and \(K\) and \(n\) two material parameters.

Equivalently, by inverting the equation, we obtain the equivalent temperature \({T}^{\mathit{eq}}\) as a function of the \({Z}_{\mathrm{\beta }}\) proportion of phase \(\mathrm{\beta }\):

(2.2)#\[\begin{split} \ {\ begin {array} {cc} {T} {T} ^ {\ mathit {eq}} ^ {T} _ {d} ^ {\ mathit {eq}} +\ frac {1} {K} {K} {\ left (\ mathrm {log}} ^ {\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left (\ frac {1}} {1} {Z} _ {\ mathrm {\ beta}}}}\ right)\ right)} ^ {1/n} &\ text {for} 0\ le {Z} _ {\ mathrm {\ beta}}\ le\ mathrm {0.99}\\ {T} ^ {\ mathit {eq}}} = {T} {eq}} = {T} _ {f}} _ {f} ^ {\ f} ^ {\ mathrm {eq}} ^ {\ mathrm {eq}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}} = {T}}} = {T}}} = {T}}} = {T}}} = {T}}} = {thrm {\ beta}}\ le 1\ end {array}\end{split}\]

The temperature at the end of transformation at equilibrium \({T}_{f}^{\mathit{eq}}\) is chosen such that corresponding to a proportion \(\mathrm{0,99}\) of phase \(\mathrm{\beta }\) transformed, i.e.:

(2.3)#\[ {T} _ {f} ^ {\ mathit {eq}}} = {T} _ {eq}} = {T} _ {d} ^ {\ mathit {eq}} +\ frac {1} {K} {\ left (\ mathrm {log}}\ left}\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\ left (\ mathrm {log}}\ left)\ left (\ mathrm {log}}\]

2.2. Evolution equation under heating#

The transformation when heated is transformation \(\mathrm{\alpha }\Rightarrow \mathrm{\beta }\).

The temperature at which phases are transformed upon heating \({T}_{c}\) depends on the temperature rate upon heating and is given by the equation:

\[\]

: label: eq-4

{T} _ {c} = {T} _ {c} {c} ^ {1} {({V} _ {mathit {ch}})}} ^ {{T} _ {c} {c} ^ {2}}

With \({V}_{\mathit{ch}}\) the heating rate in \(°C/s\) and \({T}_{c}\ge {T}_{d}^{\mathit{eq}}\).

The evolution model of phase \(\mathrm{\beta }\) during heating is given by the following differential equation (Holt model):

: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}`

\({T}^{\mathit{eq}}({Z}_{\mathrm{\beta }})\) is the equilibrium temperature corresponding to the instantaneous proportion \({Z}_{\mathrm{\beta }}\) of phase \(\mathrm{\beta }\) and given by the equation. \({T}_{c}^{1}\), \({T}_{c}^{2}\), \({A}_{c}\), \(\frac{E}{R}\), and \(M\) are material parameters.

2.3. Cooling evolution equation#

The transformation to cooling is transformation \(\mathrm{\beta }\Rightarrow \mathrm{\alpha }\).

The temperature at the beginning of phase transformation upon cooling \({T}_{r}\) depends on the temperature rate upon cooling and is given by the equation:

(2.4)#\[ {T} _ {r} = {T} _ {r} ^ {r} ^ {1} + {T} _ {r} ^ {2}\ mathrm {ln} ({V} _ {\ mathit {ref}})\]

With \({V}_{\mathit{ref}}\) the cooling rate in \(°C/s\) and \({T}_{r}\le {T}_{f}^{\mathit{eq}}\).

The evolution model from phase \(\mathrm{\beta }\) to cooling is given by the following differential equation:

: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 \({T}_{d}^{\mathit{eq}}\le T\le {T}_{f}^{\mathit{eq}}\)

\({T}_{r}^{1}\), \({T}_{r}^{2}\), \({A}_{r}\), and \({B}_{r}\) are material parameters.

2.4. Conditions of use of the metallurgical model for any temperature transients#

2.4.1. A few rules#

  • During the calculations, if the proportion of phase \(\mathrm{\beta }\) is strictly greater than \(\mathrm{0,99}\), we round to one. For a heating rate of less than \(0.1°C/s\), \({T}_{c}={T}_{d}^{\mathit{eq}}\) is used.

  • If \(0\le {Z}_{\mathrm{\beta }}\le \mathrm{0,99}\), the following rule should be applied:

    • If \(T>{T}^{\mathit{eq}}\iff {Z}_{\mathrm{\beta }}<{Z}_{\mathrm{\beta }}^{\mathit{eq}}\), apply the model to heating (even if the temperature speed is negative)

    • If \(T<{T}^{\mathit{eq}}\iff {Z}_{\mathrm{\beta }}>{Z}_{\mathrm{\beta }}^{\mathit{eq}}\), apply the model to cooling (even if the temperature rate is positive)

2.4.2. Algorithm#

We consider any transient of temperature \(T(t)\).

Note: to calculate the transformation start temperatures at heating \({T}_{c}\) and at cooling \({T}_{r}\) , 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.

Note: once the threshold temperatures \({T}_{c}\) or \({T}_{r}\) 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 \({t}_{d}^{\mathit{eq}}\) (or \({t}_{f}^{\mathit{eq}}\)) corresponding to the start temperature \({T}_{d}^{\mathit{eq}}\) (or end \({T}_{f}^{\mathit{eq}}\), respectively) of transformation at equilibrium.

  • Case where \({Z}_{\mathrm{\beta }}=0\) initially: looking for \({t}_{d}^{\mathit{eq}}\)

  • Case where \({Z}_{\mathrm{\beta }}=1\) initially: looking for \({t}_{f}^{\mathit{eq}}\)

  • Step 2: Search for the instant \({t}_{c}\) (or \({t}_{r}\)) corresponding to the transformation start temperature \({T}_{c}\) (or \({T}_{r}\), respectively) using the sliding secant method:

  • Case where \({Z}_{\mathrm{\beta }}=0\) initially: find the moment when the temperature \(T(t)\) exceeds \({T}_{c}\).

  • If \(T(t)>{T}_{d}^{\mathit{eq}}\), we increment the time, we calculate \({T}_{c}\) and we test the following condition:

(2.5)#\[ \ text {(C1)}\ mathrm {:} T (t) > {T} _ {c} > {T} _ {c} ^ {1} {\ left (\ frac {T (t) - {T} - {T} _ {d} _ {d} _ {eq}}} {\ mathit {eq}}}} {d} ^ {\ mathit {eq}}}\ right)} ^ {{T} _ {c} ^ {2}}\]

If condition \((C1)\) is true we go to step (3).

  • If \(T(t)\le {T}_{d}^{\mathit{eq}}\) without reaching \({T}_{c}\), you must then update \({t}_{d}^{\mathit{eq}}\) by restarting step 1 from the current moment.

  • Case where \({Z}_{\mathrm{\beta }}=1\) initially: find the moment when the temperature \(T(t)\) returns to \({T}_{r}\).

  • If \(T(t)<{T}_{f}^{\mathit{eq}}\), we increment the time, we calculate \({T}_{r}\) and we test the following condition:

: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)`

If condition \((C2)\) is true we go to step (3).

  • If \(T(t)\ge {T}_{f}^{\mathit{eq}}\) without reaching \({T}_{r}\), you must then update \({t}_{f}^{\mathit{eq}}\) by restarting step 1 from the current moment.

  • Step 3: Once \({T}_{c}\) (or \({T}_{r}\)) is reached, the evolution of the phase fraction \(\mathrm{\beta }\) appearing is calculated step by step using the Holt equation (on heating) or the cooling equation following the sign of \((Z-{Z}_{\mathit{eq}})\) and this as long as the phase fraction \(\mathrm{\beta }\) remains less than \(\mathrm{0,99}\) and greater than zero. and even if one goes through a temperature peak.

  • Step 4: If during step 3, the phase fraction \(\beta\) becomes equal to 1 (or 0), step 1 is repeated from the current instant.