4. Material coefficients#
In the following, the forms of evolution of the parameters of the equations are presented as a function of the control variables temperature and fluence. The values of these parameters for materials 304 and 316 are recorded in the HT-26/05/045/A report.
4.1. Thermo-elasticity#
Young’s modulus [ref] is given by:
\(E={C}_{0}^{E}+{C}_{1}^{E}\cdot T\) eq 4.1-1
The Poisson’s ratio is given by:
\(\nu ={C}_{0}^{\nu }+{C}_{1}^{\nu }\cdot T\) eq 4.1-2
The thermal tangent expansion coefficient is given by:
\(\alpha ={C}_{0}^{\alpha }+{C}_{1}\alpha \cdot T+{C}_{2}^{\alpha }\cdot {T}^{2}\) eq 4.1-3
The secant expansion coefficient is then given, taking a zero reference temperature, by:
\({\alpha }^{\text{sec}}={C}_{0}^{\alpha }+\frac{1}{2}{C}_{1}^{\alpha }\cdot T+\frac{1}{3}{C}_{2}^{\alpha }\cdot {T}^{2}\) eq 4.1-4
4.2. Plasticity#
This part uses the [ref] ratio to calculate the work-hardening curves of the steels constituting the internal structures of tanks after irradiation for various temperatures. We use the expressions of the elastic limit at 0.2% of plastic deformation \({R}_{\mathrm{0,2}}\), of the ultimate stress \({R}_{m}\) and of the distributed elongation \({e}_{u}\) as a function of temperature \(T\), irradiation \(\Phi\) (the [ref] ratio uses the term \(d\) to describe irradiation) and of the cold work hardening rate \(c\). The [ref] ratio also provides the elongation at break but this data cannot be used in practice because it depends on the type of test piece studied.
The distributed elongation is expressed as:
\({e}_{u}={e}_{u}^{0}(T){\eta }_{3}(c){\xi }_{3}(d)\) eq 4.2-1
The elastic limit at 0.2% is expressed as:
\({R}_{\mathrm{0,2}}={R}_{\mathrm{0,2}}^{0}(T){\eta }_{1}(c){\xi }_{1}(d)\) eq 4.2-2
The ultimate step is not expressed directly. First of all, difference \(\Delta R={R}_{m}-{R}_{\mathrm{0,2}}\) is represented by a function:
\(\Delta R=({R}_{m}^{0}(T)-{R}_{\mathrm{0,2}}^{0}(T)){\eta }_{2}(c){\xi }_{2}(d)\) eq 4.2-3
Adjusting \(\Delta R\) instead of \({R}_{m}\) ensures that: \({R}_{m}>{R}_{\mathrm{0,2}}\). So \({R}_{m}\) is obtained as: \({R}_{m}(T,c,d)={R}_{\mathrm{0,2}}(T,c,d)+\Delta R(T,c,d)\)
Note that the functions \({e}_{u}^{0}\), \({R}_{\mathrm{0,2}}^{0}\) and \({R}_{m}^{0}\) are valid for the two materials (304 and 316) that make up tank internals. Functions \({\eta }_{\mathrm{1,2}\mathrm{,3}}\) and \({\xi }_{\mathrm{1,2}\mathrm{,3}}\) depend on the material. It is proposed to represent the hardening curve of materials for values of \(T\), \(d\) and \(c\) given by a power law of the type:
\({\sigma }_{f}(p)=K{(p+{p}_{0})}^{n}\) eq 4.2-4
where \(p\) is the equivalent Von-Misès plastic deformation. \(K\), \({p}_{0}\), and*n* are parameters to be calculated in order to obtain the values of \({R}_{\mathrm{0,2}}^{0}\), \({R}_{m}\), and \({e}_{u}\).
The value of the deformation corresponding to the distributed elongation (noted \({\varepsilon }_{u}=\mathrm{log}(1+{e}_{u})\)) is obtained by the Consider condition (neglecting the elastic deformation):
\(\frac{d{\sigma }^{f}}{\text{dp}}={\sigma }^{f}\) eq 4.2-5
either:
\(nK{(p+{p}_{0})}^{n-1}=K{(p+{p}_{0})}^{n}\Rightarrow {p}^{0}=n-{\varepsilon }_{u}\) eq 4.2-6
The ultimate constraint is equal to:
\({R}_{m}={\sigma }^{f}({\varepsilon }_{u})\mathrm{exp}(-{\varepsilon }_{u})=K{n}^{n}\mathrm{exp}(-{\varepsilon }_{u})\) eq 4.2-7
Either:
\(K=\frac{{R}_{m}}{{n}^{n}}\mathrm{exp}({\varepsilon }_{u})\) eq 4.2-8
The elastic limit is given by (the variation in cross section is neglected here):
\({R}_{\mathrm{0,2}}=K{({p}_{e}+{p}_{0})}^{n}\) with \({p}_{e}=\mathrm{0,002}\) eq 4.2-9
It therefore remains to solve a single nonlinear equation with respect to n:
\(S={R}_{\mathrm{0,2}}-\frac{{R}_{m}}{{n}^{n}}\mathrm{exp}({\varepsilon }_{u}){({p}_{e}+n-{\varepsilon }_{u})}^{n}=0\) eq 4.2-10
The search for the solution is done by dichotomy by taking \({\varepsilon }_{u}\) as the initial value of \(n\). We then calculate \(n\) and \(K\) using the equations and.
In the case where the material has little work-hardening (i.e. high irradiation or very significant cold work hardening), it is not possible to find a solution to the equation. In this case we will use the power law of the form \({\sigma }^{f}(p)=K{p}^{n}\) with \(n={\varepsilon }_{u}\) and \({\sigma }^{f}(p)=K{p}^{\mathrm{nK}}={R}_{m}\mathrm{exp}({\varepsilon }_{u})/{n}^{n}\) being \({p}_{0}=0\).
Using the previous laws for low p values can lead to unrealistic results. It can be considered that the flow stress cannot be less than \(\kappa {R}_{\mathrm{0,2}}\) (with \(\kappa\) close to 1). The effect of choosing \(\kappa\) on the creep response of a structure is negligible for values between 0.8 and 1. It is recommended, for a calculation in Code_Aster, to use the value \(\kappa =\mathrm{0,8}\). In addition, a linear extrapolation between \(p=0\) and \(p={p}_{e}\) obtained from the values in \({p}_{e}\) of the stress and the work hardening will be used. The figure shows schematically the shape of the proposed work-hardening law.
Figure 4.2-1: Proposed work hardening law.
4.3. Irradiation creep#
The irradiation creep data collected in [ref] makes it possible to determine the values of \({A}_{i}\) and \({\eta }_{i}^{s}\). These values are in principle constant with temperature. However, it is likely that at low temperatures, there will be no radiation creep. To model this evolution, it is possible to make the coefficient \({A}_{i}\) depend on temperature:
\({A}_{i}={A}_{i}^{0}{\zeta }_{f}(T)\) eq 4.3-1
where \({A}_{0}^{i}\) is the parameter value for high temperatures. In this equation \({\zeta }_{f}\) is a function that can make it possible to stop the phenomenon of creep below a temperature threshold. It can be written in the form \({\zeta }_{f}(T)=\frac{1}{2}(1+\text{tanh}({\mu }_{T}(T-{T}_{c})))\), where \({T}_{c}\) makes it possible to adjust the temperature at which the irradiation creep starts and where \({\mu }_{T}\) makes it possible to adjust the width of the transition between the temperature domains with and without irradiation creep.
4.4. Swelling#
We will use a bilinear Foster’s law which makes it possible to represent an incubation time and then a linear swelling [ref, ref]. We then have:
\(\frac{\Delta V}{{V}_{0}}={F}_{g}(\Phi )=R\cdot (\Phi +\frac{1}{\alpha }\text{Log}(\frac{1+\mathrm{exp}(\alpha ({\Phi }_{0}-\Phi ))}{1+\mathrm{exp}(\alpha {\Phi }_{0})}))\) eq 4.4-1
For \(\Phi \to \infty\) we get:
\(\frac{\Delta V}{{V}_{0}}=R\cdot \Phi -\frac{R}{\alpha }\text{Log}(1+\mathrm{exp}(\alpha {\Phi }_{0}))\) eq 4.4-2
The incubation fluence is therefore: \(\text{Log}(1+\mathrm{exp}(\alpha {\Phi }_{0}))/\alpha\) (note: if \(\alpha {\Phi }_{0}\gg 1\) it is equal to \({\Phi }_{0}\)). By deriving the equation we get:
\({A}_{g}=\frac{1}{3}R(1-\frac{\mathrm{exp}(\alpha ({\Phi }_{0}-\Phi ))}{1+\mathrm{exp}(\alpha ({\Phi }_{0}-\Phi ))})\) eq 4.4-3
Figure 4.4-1: Foster’s bilinear law [ref].
The material parameters of the swelling law are therefore R, \(\alpha\), and \({\Phi }_{0}\).
As in the case of radiation creep, it may be necessary to introduce a dependence on temperature in order not to impose swelling at low temperature. It is possible to make \(R\) depend on temperature by using a function \({\zeta }_{g}\) similar to that used for irradiation creep:
\(R={R}^{0}{\zeta }_{g}(T)\) with \({\zeta }_{g}(T)=\frac{1}{2}(1+\mathrm{tanh}({\mu }_{g}(T-{T}_{c}^{g})))\) eq 4.4-4