Subsequently, the equations of the model are presented in a « generic » coordinate system (1,2,3) which represents either the Cartesian coordinate system (Ox, Oy, Oz), or the cylindrical coordinate system \((1\mathrm{=}{e}_{r},2\mathrm{=}{e}_{\theta }\mathrm{,3}\mathrm{=}z)\) associated with the shaft sheath \(z\).
4.2. Model equations
The deformations are divided into elastic \({\mathrm{\epsilon }}^{e}\), thermal \({\mathrm{\epsilon }}^{\mathit{th}}\) and viscous \({\mathrm{\epsilon }}^{v}\) parts:
(4.1)\[ \ mathrm {\ epsilon} = {\ mathrm {\ epsilon}}} ^ {e} + {\ mathrm {\ epsilon}} ^ {\ mathit {th}}\ mathit {th}}\ mathit {Id} + {\ mathrm {\ epsilon}} ^ {v}}\]
For the stress-deformation relationship: we separate the deviatoric part from the spherical part:
(4.2)\[ \ mathrm {\ sigma} =\ stackrel {~} {\ mathrm {\ sigma}} +\ frac {1} {3} {\ mathrm {\ sigma}}} _ {\ mathit {pp}} {\ mathit {pp}}}\ mathit {Id}\]
And we have:
\[\]
: label: eq-4
stackrel {~} {mathrm {sigma}}} =2mathrm {mu}left (stackrel {~} {mathrm {epsilon}}} - {mathrm {epsilon}}} - {mathrm {epsilon}}} ^ {v}right)
With the flow law of viscous deformation such as:
\[\]
: label: eq-5
{dot {mathrm {epsilon}}}} ^ {v}} =dot {p}frac {Mmathrm {:}mathrm {sigma}}} {{mathrm {sigma}}} _ {mathit {eq}}}
with the equivalent stress in the Hill sense defined by:
(4.3)\[ {\ mathrm {\ sigma}} _ {\ mathit {eq}} =\ sqrt {\ mathrm {\ sigma}\ mathrm {:} M\ mathrm {:}\ mathrm {:}\ mathrm {\ sigma}}\]
The Hill anisotropy matrix, \(M\), is of the form:
(4.4)\[\begin{split} {M} _ {\ left ({e} _ {r}, {r}, {e} _ {\ mathrm {\ theta}}}, {e} _ {z}\ right)} =\ left [\ begin {array} {cccc} {{r}} {r}} {r}} {r} {r}} {r} {r}} {r} {r}}, {r} _ {r} _ {r} _ {12}} & {M} _ {22} & {M} _ {23} & 0& 0& 0& 0& 0& 0\\ {M} _ {23} & {M} _ {33} & 0& 0& 0\ & 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0} & 0\ 0& 0& 0& {M} _ {55} & 0\\ 0& 0& 0& 0& 0& {M} _ {66}\ end {array}\ right]\end{split}\]
With:
(4.5)\[ \ {\ begin {array} {c} {M} _ {22} + {M} _ {11} + {M} _ {12} + {M} _ {12} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M} _ {22} + {M}\]
In the isotropic case, we have:
(4.6)\[ {M} _ {11} = {M} _ {22} = {M} _ {33} =1
{M} _ {12} = {M} _ {13} = {M} _ {23} =-\ frac {1} {2}
{M} _ {44} = {M} _ {55} = {M} _ {66} =\ frac {3} {4}\]
The terms of this matrix depend on the phase distribution, with:
(4.7)\[\begin{split} M=\ {\ begin {array} {c} {M} {M} ^ {3}\ text {si} 3}\ text {si} 0\ le {\ begin {array}}\ le\ mathrm {0.01}\ hfill\\ {M} ^ {3}}\ text {si} {3}\ text {si} 0\ text {si} 0\ le {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} _ {2} = {Z} mathrm {\ alpha}}) {M} ^ {3}\ text {si}\ text {si}\ mathrm {0.01}\ le {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.99}\ hfill\\ {3}\ hfill\\ {M}}\ hfill\\ {M} ^ {M} ^ {1}\ text {\ alpha}\ the {Z}}}\ le {Z} _ {\ mathrm {0.99}\ the {Z} _ {\ mathrm {0.99}}\ the {Z} _ {\ mathrm {0.99}\ hfill\\ {M}}\ hfill\\ {M} ^ {M} ^ {1} ^ {1}\ text {\ alpha}\ the {Z}}\ le 1\ hfill\ end {array}\end{split}\]
The equivalent deformation rate is given by:
(4.8)\[ \ dot {p} = {\ left (\ frac {{\ mathrm {\ sigma}}} _ {\ mathit {eq}}}} {{\ mathit {ap}}} ^ {m}}\ right)}}\ right)} ^ {n} {e} ^ {-Q/T}\]
Or equivalently:
(4.9)\[ {\ mathrm {\ sigma}} _ {\ mathit {eq}}} =a {({e} ^ {Q/T})} ^ {1/n} {p} ^ {m} {\ dot {p}} {\ dot {p}}}} ^ {p}}} ^ {p}}} ^ {p}} {p} {m} {\ dot {p}}} {\ dot {p}}} {\ dot {p}}}} {\ dot {p}}}} {\ dot {p}}}} {\ dot {p}}}} {\ dot {p}}}} {\]
We apply the law of mixtures on viscous stress \({\mathrm{\sigma }}_{v}\):
(4.10)\[ {\ mathrm {\ sigma}} _ {\ mathit {eq}}} = {\ mathrm {\ sigma}} _ {v} =\ sum _ {i=1} ^ {3} {f} _ {i} _ {i} ({Z}}} ({Z}} _ {Z} _ {Z} _ {z} _ {v, i}) {\ mathrm {\ sigma}} _ {v, i}\]
With:
(4.11)\[\begin{split} {f} _ {1} =\ {\ begin {array} {c} 0\ text {si} 0\ text {si} 0\ le {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.9}\ hfill\\ fill\\\ frac {\ frac {{Z}} _ {\ mathrm {\ alpha}}}\ text {si}\ mathrm {0.9}\ le {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.99}\ hfill\\ 1\ text {si}\ mathrm {0.99}\ le {Z} _ {\ mathrm {\ alpha}}\ le 1\ hfill\ end {array}\ le 1\ hfill\ end {array}
{f} _ {3} =\ {\ begin {array} {c} 0\ text {si} 0\ the {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.01}\ hfill\\\ frac {\\ frac {\\ frac {\ mathrm {0.1}} - {Z} _ {\ mathrm {\ alpha}}} {\ mathrm {0.01}}\ hfill\\\ frac {\\ frac {\ frac {\ frac {\ frac {0.1} - {Z} _ {\ mathrm {\ alpha}}}} {\ mathrm {0.09}}\ text {si}\ mathrm {0.01}\ le {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.1}\ hfill\\ 1\ text {si}\ mathrm {0.1}\ mathrm {0.1}\ le {Z} _ {\ mathrm {\ alpha}}\ le 1\ hfill\ end {array}\
{f} _ {2} =\ {\ begin {array} {c} 0\ text {si} 0\ the {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.01}\ hfill\\ 1-\ frac {\ mathrm {0.1}} - {Z} _ {\ mathrm {\ alpha}}} {\ mathrm {0.01}}\ hfill\\ 1-\ frac {\ frac {\ frac {0.1}} - {Z} _ {\ mathrm {\ alpha}}} {\ mathrm {0.01}}\ hfill\\ 1-\ frac {\ frac {\ frac {0.1}} - {Z} _ {\ mathrm {\ alpha}}} {\ mathrm {0.01}}\ hfill\ text {si}\ mathrm {0.01}\ le {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {0.1}\ hfill\\ 1\ text {si}\ mathrm {0.1}\ mathrm {0.1}\ le {Z}\ le {Z} _ {\ mathrm {\ alpha}}\ le\ mathrm {\ alpha}}\ le\ mathrm {\ alpha}}\ le\ mathrm {\ alpha}}\ le\ mathrm {\ alpha}}\ le\ mathrm {\ alpha}}\ le\ mathrm {\ 1-\ frac {{Z} _ {\ mathrm {\ alpha}} -\ mathrm {0.9}} {\ mathrm {0.9}}\ text {si}\ mathrm {0.9}\ the {Z} _ {\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the\ mathrm {\ alpha}}\ the {\ alpha}}\ le 1\ hfill\ end {array}\end{split}\]
where \(\left({a}_{i},{Q}_{i},{n}_{i},{m}_{i}\right)\) are material parameters associated with the three metallurgical phases.
