2. Local integration of Rankine’s law
The plastic deformation rate is given using the Koiter formula:
(2.1)\[ d {\ mathrm {\ epsilon}}} ^ {p} =\ sum _ {j=1} ^ {m} d {\ mathrm {\ mu}} _ {j}\ frac {\ partial {R} _ {R} _ {j}} _ {j}}} =\ sum _ {j=1}} ^ {m} d {\ mathrm {\ mu}} _ {j} {n} _ {j}\]
Where \(d{\mathrm{\mu }}_{i}\ge 0\) are the plastic multipliers associated with the \(i\) mechanisms, and:
(2.2)\[ \ frac {\ partial {R} _ {i}}} {\ partial {\ mathrm {\ sigma}} _ {j}} = {n} _ {\ mathit {ij}}} = {\ mathrm {\ delta}}} = {\ mathrm {\ delta}} _ {\ mathit {ij}}\]
And where \(m\) characterizes the number of active mechanisms, equal to one, two or three depending on the following situations:
the final stress is located inside the load surface, the point is regular and \(m=1\);
the final constraint is located on an edge of the cone, the point is singular and \(m=2\);
the final stress is not located inside the load surface or on an edge. It is then projected to the top of the cone, the point is singular and \(m=3\);
The final stress \({\sigma }^{\text{+}}\) is calculated on the basis of an elastic prediction noted \({\mathrm{\sigma }}^{e}\) and a correction \(\mathrm{\Delta }{\mathrm{\sigma }}_{C}=C\mathrm{.}\mathrm{\Delta }{\mathrm{\epsilon }}^{p}\) so that:
\[\]
: label: eq-4
{mathrm {sigma}} ^ {text {+}}} = {mathrm {sigma}} ^ {e} -mathrm {Delta} {mathrm {sigma}}} _ {C}} _ {C}} = {mathrm {+}}} = {mathrm {sigma}}} ^ {e} -mathrm {sigma}}} ^ {e} -mathrm {sigma}} _ {sigma}} _ {sigma}} _ {C} = {mathrm {sigma}} = {mathrm {sigma}}} ^ {e} -sum _ {j=1} ^ {m}mathrm {sigma}} _ {sigma}} _ {sigma}} {mu}} _ {j} Cmathrm {.} {n} _ {j}
Plastic multipliers \(d{\mathrm{\mu }}_{j}\) are calculated by injecting equation () into equation (), which gives:
\[\]
: label: eq-5
sum _ {j=1} ^ {m} d {mathrm {mu}} _ {j} {left (Cmathrm {.} {n} _ {j}right)} _ {i} = {mathrm {sigma}} _ {i} ^ {e} - {mathrm {sigma}}} _ {t}
In what follows, the expressions corresponding to the various situations mentioned above are detailed.
Volume and equivalent plastic deformations are written as:
(2.3)\[\begin{split} \ {\ begin {array} {c} {\ mathrm {\ epsilon}}} _ {v} ^ {p} = {\ mathrm {\ epsilon}} _ {1} + {\ mathrm {\ epsilon}} + {\ mathrm {\ epsilon}}} _ {3}\\ {\ stackrel {~} {epsilon}} _ {\ stackrel {~} {epsilon}} _ {\ mathrm {\ epsilon}} _ {\ stackrel {~} {epsilon}} _ {\ mathrm {\ epsilon}} m {\ epsilon}}} ^ {p} =\ green {\ stackrel {~} {\ mathrm {\ epsilon}}}} ^ {p}\ green =\ sqrt {\ frac {2} {2} {3} {3} {\ stackrel {3}} {\ stackrel {~} {\ stackrel {~} {\ stackrel {~} {\ stackrel}}}} ^ {p}\ mathrm {:} {\ stackrel {\ stackrel {~}} {\ stackrel {~}} {\ stackrel {~}} {\ stackrel}}} ^ {p}\ mathrm {:} {\ stackrel {\ stackrel {~}} {\ stackrel {~}} ~} {\ mathrm {\ epsilon}}}}} ^ {p}}\ end {array}\end{split}\]
With \({\stackrel{~}{\mathrm{\epsilon }}}^{p}={\mathrm{\epsilon }}^{p}-\frac{{\mathrm{\epsilon }}_{v}^{p}}{3}1\).
2.1. Cases where only one mechanism is active
The \({R}_{1}\) mechanism is active. From where:
(2.4)\[ d\ mathrm {\ mu}\ left (K+\ frac {4} {4} {4} {3} G\ right) = {R} _ {1}\ left ({\ mathrm {\ sigma}}} ^ {e}}\ right) = {\ mathrm {\ e}} = {\ mathrm {\ sigma}}\ right) = {\ mathrm {\ e}\ right) = {\ mathrm {\ e}\ right) = {\ mathrm {\ e}\ right) = {\ mathrm {\ e}\ right) = {\ mathrm {\ e}\ right) = {\ mathrm {\ e}\ right) = {\ mathrm {\ e}\ right)\]
With \({\mathrm{\sigma }}_{t}\) the material’s tensile limit. From where:
(2.5)\[ d\ mathrm {\ mu} =\ frac {{{R} _ {1} ({\ mathrm {\ sigma}} ^ {e}) ⟩}} _ {\ text {+}}}} {K+\ frac {4} {3} G}}\]
Where the operator \({⟨\text{}⟩}_{\text{+}}\) designates the positive part of the associated quantity.
Plastic deformation is written as:
(2.6)\[\begin{split} \ mathrm {\ delta} {\ mathrm {\ epsilon}}} ^ {p}} =\ mathrm {\ delta}\ mathrm {\ mu}\ frac {\ partial {R} _ {1}}} {\ partial\ mathrm {\ epsilon}}}} {\ partial\ epsilon}}} ^ {p} =\ mathrm {\ delta}\ mathrm {\ mu}\ frac {\ partial {R} _ {1}}}} {\ mathrm {\ epsilon}}} {\ mathrm {\ epsilon}}} {\ mathrm {\ epsilon}}} =\ mathrm {\ delta}\ mathrm {\ mu}\ frac {\ partial {R} _ {1}} 1\\ 0\\ 0\ end {array}\ right)\end{split}\]
And correction \(\mathrm{\Delta }{\mathrm{\sigma }}_{C}\) reads:
(2.7)\[\begin{split} \ mathrm {\ Delta} {\ mathrm {\ sigma}} _ {C} =C\ mathrm {.} \ mathrm {\ delta} {\ mathrm {\ epsilon}}} ^ {p} =\ mathrm {\ delta}\ mathrm {\ mu} C\ mathrm {.} \ left (\ begin {array} {c} 1\\ 0\\ 0\\ 0\ end {array}\ right) =\ mathrm {\ delta}\ mathrm {\ mu}\ left (\ begin {array} {c} {c} K+\ frac {4}} {3} K+\ frac {4} {4} {3} {3} {3} G\ end {array}\ right)\end{split}\]
Volume and equivalent plastic deformations are written as:
(2.8)\[\begin{split} \ {\ begin {array} {c}\ mathrm {\ delta} {\ delta} {\ mathrm {\ epsilon}} _ {v} ^ {p} =\ mathrm {\ delta}\ mathrm {\ mu}\\ {\ delta}\\ mathrm {\ delta}\ stackrel {\ delta}}} {\ mathrm {\ epsilon}}} ^ {p} =\ sqrt {p} =\ sqrt {\ delta}}\ frac {2} {3}\ mathrm {.} \ frac {1} {9}\ left (\ begin {array} {c} {c} 2\ mathrm {\ delta}\ mathrm {\ mu}\\ -\ mathrm {\ delta}\ mathrm {\ mu}\\ -\ mathrm {\ mu}\ mathrm {\ delta}\ mathrm {\ mu}\ end {array}\ delta}\ right)\ mathrm {\ mu}\ right)\ mathrm {.} \ left (\ begin {array} {c} 2\ mathrm {\ delta}\ mathrm {\ delta}\\ -\ mathrm {\ delta}\ mathrm {\ mu}\\ -\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ thrm {\ mu}\ end {array}\end{split}\]
2.2. Cases where two mechanisms are active
Mechanisms \({R}_{1}\) and \({R}_{2}\) are active. From where:
(2.9)\[ \ {\ begin {array} {c} d {\ mathrm {\ mu}}} _ {1}\ underset {A} {\ underset {} {\ left (K+\ frac {4} {3} {3} G\ right)}}} +d {\ mathrm {\ mu}}} _ {2}\ underset {B} {\ underset {} {\ left} {\ left (K-\ right)}} +d {\ mathrm {\ mu}}} _ {2}\ underset {B} {\ left {} {\ left (K-\ right)}} frac {2} {3} G\ right)}}} = {R} _ {1}}\ left ({\ mathrm {\ sigma}} ^ {e}\ right) = {\ mathrm {\ sigma}}} _ {1} ^ {e}} _ {1} {e} - {\ mathrm {\ sigma}} - {\ mathrm {\ sigma}} _ {1} {e}} - {\ mathrm {\ sigma}} _ {1} {e}} - {\ mathrm {\ sigma}} _ {1}\ left (K-\ frac {2} {3} G\ right) +d {\ mathrm {\ mu}}} _ {2}\ left (K+\ frac {4} {3} G\ right) = {R} _ {2}\ right) = {R} _ {2}\ right) = {\ mathrm {\ sigma}} _ {2}\ left ({\ mathrm {\ sigma}}} _ {2}\ left ({\ mathrm {\ sigma}}} _ {2}\ left ({\ mathrm {\ sigma}}} _ {2}\ left ({\ mathrm {\ sigma}}} e} - {\ mathrm {\ sigma}}} _ {t}\ end {array}\]
Assuming \(\mathit{det}\) as the determinant of the system, we have:
(2.10)\[ \ mathit {det} = {A} ^ {2} - {B} ^ {2} =4G\ left (K+\ frac {G} {3}\ right) >0\]
Since the determinant is always strictly zero, there is always a solution that can be written as:
(2.11)\[\begin{split} \ {\ begin {array} {c} d {\ mathrm {\ sigma}} {\ mathrm {\ mu}}}} _ {1}\ left ({\ mathrm {\ sigma}}} ^ {e}\ right}} ^ {e}\ right) - {\ mathit {\ mu}}}\ right) - {\ mathit {\ mu}}\ right) - {\ mathit {BR}} _ {2}\ left ({\ mathrm {\ sigma}}\ left ({\ mathrm {\ sigma}}} {\ sigma}}} ^ {e}\ right) - {\ mathit {\ mu}}\ right) - {\ mathit {BR}}} _ {2}\ left ({\ mathrm {\ sigma}}} {\ sigma} right) ⟩} _ {\ text {+}}}} {\ mathit {det}}} {\ mathit {det}}}\\ d {\ mathrm {\ mu}} _ {2}}} {\ mathit {AR}}}} _ {2}\ left ({\ mathrm {\ sigma}}}\ left ({\ sigma}}} ^ {e}\ right) - {\ mathit {BR}}} _ {1}\ left ({\ left) mathrm {\ sigma}} ^ {e}\ right) ⟩} __ {\ text {+}}} {\ mathit {det}}}\ end {array}}\ end {array}\end{split}\]
Plastic deformation is written as:
(2.12)\[\begin{split} \ mathrm {\ delta} {\ mathrm {\ epsilon}}} ^ {P}} =\ mathrm {\ delta} {\ mathrm {\ mu}} _ {1}\ frac {\ partial {R} _ {R} _ {R} _ {1}} _ {2}} {\ partial\ mathrm {\ mu}} _ {2}\ frac {\ partial {R} _ {2}}} {\ partial\ mathrm {\ sigma}} =\ mathrm {\ delta} {\ mathrm {\ mu}} _ {1}\ left (\ begin {array} {2}}}} {\ left (\ begin {array} {2}}}} {\ left) (\ begin {array} {2}}}} {\ left (\ begin {array} {2}}}} {\ left (\ begin {array} {2}}}} {\ left (\ begin {array} {2}}}} {\ left (\ begin {array} {2}}}} {\ left) (\ begin {array} {2}}}} {\ left (\ begin {array} {2}}}} {\ left (_ {2}\ left (\ begin {array} {c} {c} 0\\ 1\\ 0\ end {array}\ right)\end{split}\]
And correction \(\mathrm{\Delta }{\mathrm{\sigma }}_{C}\) reads:
(2.13)\[\begin{split} \ mathrm {\ Delta} {\ mathrm {\ sigma}} _ {C} =C\ mathrm {.} \ mathrm {\ delta} {\ mathrm {\ epsilon}}} ^ {P} =\ mathrm {\ delta} {\ mathrm {\ mu}}} _ {1} C\ mathrm {.} \ left (\ begin {array} {c} 1\\ 0\\ 0\\ 0\ end {array}\ right) +\ mathrm {\ delta} {\ mathrm {\ mu}}} _ {2} C\ mathrm {.} \ left (\ begin {array} {c} 0\\ 1\\ 1\\ 0\ end {array}\ right) =\ mathrm {\ delta} {\ mathrm {\ mu}} _ {1}\ left (\ begin {array} {c} {c} K+\ frac {c} {c} K+\ frac {4} {3} K+\ frac {4} {3} K+\ frac {4} {3} G\\ K-\ frac {2} {3} G\\ K-\ frac {2} {3} G\\ K-\ frac {2}} {3} G\ end {array}\ right) +\ mathrm {\ delta} {\ mathrm {\ mu}} _ {2}\ left (\ begin {array} {c} K-\ frac {2} {2} {2} {2} {2} {2} {3} K-\ frac {2} {3} K-\ frac {2} {3} K-\ frac {2} {2} {3} G\ frac {2} {2} {3} G\ frac {2} {2} {3} G\ frac {2} {2} {3} G\ frac {2} {2} {3} G\ frac {2} {3} G\ frac {2} {3} G\ frac {2} {3}\end{split}\]
Volume and equivalent plastic deformations are written as:
(2.14)\[\begin{split} \ {\ begin {array} {c}\ mathrm {\ delta} {\ delta} {\ mathrm {\ epsilon}} _ {v} ^ {P} =\ mathrm {\ delta} {\ delta} {\ mathrm {\ mu}} {\ mathrm {\ mu}} {\ delta}} =\\ {\ delta} {\ mathrm {\ mu}} {\ mathrm {\ mu}} {\ mathrm {\ mu}}} _ {2}\\ {\ mathrm {\ mu}} {\ mathrm {\ mu}}} _ {2}\\ {\ mathrm {\ mu}}}\ stackrel {~} {\ mathrm {\ epsilon}}}} ^ {p} =\ sqrt {\ frac {2} {3}\ mathrm {.} \ frac {1} {9}\ left (\ begin {array} {c} {c} 2\ mathrm {\ delta}} {\ mathrm {\ mu}} _ {1} -\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ delta}} - {2}\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}} _ {2} -\ mathrm {\ mu}} _ {2} -\ mathrm {\ mu}} m {\ delta} {\ mathrm {\ mu}}} _ {1}\\ -\ mathrm {\ delta} {\ mathrm {\ mu}} _ {1} -\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ {} {\ mathrm {\ mu}}} _ {2}\ end {array}\ right)\ mathrm {.} \ left (\ begin {array} {c} 2\ mathrm {\ delta} {\ delta} {\ mathrm {\ mu}}} _ {1} -\ mathrm {\ delta} {\ mu}} {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}} {\ delta}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ mathrm {\ mu}}} _ {2} -\ m {\ mu}} _ {1}\\ -\ mathrm {\ delta} {\ mathrm {\ delta}} _ {1} -\ mathrm {\ delta} {\ mathrm {\ mu}}} _ {2}\ mathrm {\ mu}}}} _ {2}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}}} _ {\ mathrm {\ mu}} _ {1} ^ {2} +\ mathrm {\ delta} {\ mathrm {\ mu}} _ {2} ^ {2} -\ mathrm {\ delta} {\ mathrm {\ mu}} +\ mathrm {\ mu}} +\ mathrm {\ delta}} _ {\ delta}} _ {2}} {\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ delta}} -\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta}\end{split}\]
2.3. Case of the projection at the top of the cone
In this case, we have directly:
(2.15)\[\begin{split} \ {\ begin {array} {c} {p} ^ {p} ^ {\ text {+}} = {p} ^ {e} -K\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} _ {v} ^ {p}} ^ {p} = {\ mathrm {\ sigma}}} _ {t}\\ {\ mathrm {\ sigma}}} ^ {\ text {+}} = {\ text {+}} = {\ mathrm {\ sigma}}} ^ {\ text {+}} = {p} ^ {\ text {+}}} 1\ end {array}\end{split}\]
Volume plastic deformation is obtained directly:
(2.16)\[ \ mathrm {\ Delta} {\ mathrm {\ epsilon}}} _ {v} ^ {p} =\ frac {{p} ^ {e} - {\ mathrm {\ sigma}}} _ {t}} {K}\]
Mechanisms \({R}_{1}\), \({R}_{2}\), and \({R}_{3}\) are active. From where:
(2.17)\[\begin{split} \ {\ begin {array} {c} d {\ mathrm {\ mu}}} _ {1} A+\ left (d {\ mathrm {\ mu}} _ {2} +d {\ mathrm {\ mu}}} _ {3}}}} _ {3}\ right) _ {3}\ right) B= {R} _ {1}\ right) = {\ mathrm {\ sigma}} _ {1} ^ {e} - {\ mathrm {\ sigma}} - {\ mathrm {\ sigma}} _ {2} A+\ left (d {\ mathrm {\ mu}} - {1}}} _ {1}} +d {\ mathrm {\ mu}}} _ {3}\ right) B= {R}\ left (d {\ mathrm {\ mu}}} _ {3}\ right) B= {R}\ left (d {\ mathrm {\ mu}}} _ {3}\ right) B= {R}\ left (d {\ mathrm {\ mu}}} _ {3}\ left) B= {R}\ left (d {\ mathrm {\ mu}}} 2}\ left ({\ mathrm {\ sigma}}} ^ {e}\ right) = {\ mathrm {\ sigma}} _ {2} ^ {e} - {\ mathrm {\ sigma}} _ {t} _ {t}\\ d {\ d}\\ d {\ mathrm {\ mu}}} _ {1} +d {\ mathrm {\ mu}} _ {2}\ right) B= {R} _ {3}\ left ({\ mathrm {\ sigma}} ^ {e}\ right) = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}}\ right) = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}\ right) = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}} = {\ mathrm {\ sigma}\end{split}\]
We show that after some algebraic manipulations, we obtain:
\[\]
: label: eq-21
{begin {array} {c} d {mathrm {mu}}} _ {1}} =frac {left (A+Bright) {R} _ {1} -Bleft ({R} _ {2}} + {R} _ {2}} + {R} _ {2} + {R} _ {2})} _ {2}} =frac {left (A+Bright) {R} _ {2} =frac {left (A+Bright) {R} _ {2} -Bleft ({R} _ {1} + {R} _ {3}right)} {6mathit {KG}}}\d {mathrm {mthrm {mu}}} _ {3}} _ {3} -Bleft ({R} _ {1} + {R} _ {2}right)} {6mathit {KG}}end {array}
The equivalent plastic deformation is then written as:
(2.18)\[\begin{split} {\ mathrm {\ delta} e} ^ {P} =\ sqrt {\ frac {2} {3}\ mathrm {.} \ frac {1} {9}\ left (\ begin {array} {c} {c} 2\ mathrm {\ delta} {\ mathrm {\ mu}} _ {1} -\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ delta}} -\ mathrm {\ mu}} -\ mathrm {\ mu}} _ {3}\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} m {\ delta} {\ mathrm {\ mu}} _ {2} -\ mathrm {\ delta} {\ mathrm {\ mu}} _ {1} -\ mathrm {\ delta} {\ delta} {\ mathrm {\ mu}} {\ mathrm {\ mu}} {\ mathrm {\ mu}} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ {\ delta} {\ mathrm {\ mu}}} _ {1} -\ mathrm {\ delta} {\ mathrm {\ mu}} _ {2}\ end {array}\ right)\ mathrm {.} \ left (\ begin {array} {c} 2\ mathrm {\ delta} {\ delta} {\ mathrm {\ mu}}} _ {1} -\ mathrm {\ delta} {\ mu}}} _ {2} -\ mathrm {\ delta} {\ delta} {\ delta} {\ delta} {\ mathrm {\ mu}}} _ {3}\\ 2\ mathrm {\ mu}} _ {\ delta}} _ {3}\\ 2\ mathrm {\ mu}} _ {\ delta} {\ mathrm {\ mu}} m {\ mu}} _ {2} -\ mathrm {\ delta} {\ mathrm {\ mu}}} _ {1} -\ mathrm {\ delta} {\ mathrm {\ mu}}} _ {3}} _ {3}} _ {3}} _ {3} -\ mathrm {\ mu}}} _ {3} -\ mathrm {\ delta}} _ {3} -\ mathrm {\ delta}} _ {3} -\ mathrm {\ mu}}} _ {3} -\ mathrm {\ mu}}} _ {3} -\ mathrm {\ mu}}} _ {3} -\ mathrm {\ mu}}} _ {3} -\ mathrm {\ mu}}} _ {3} {\ mu}} _ {1} -\ mathrm {\ delta} {\ mathrm {\ mu}} _ {2}\ end {array}\ right)}
{\ mathrm {\ delta}\ stackrel {~} {\ mathrm {\ epsilon}}}} ^ {p} =\ frac {2} {3}\ sqrt {\ delta {\ mu} _ {1} {\ mu} _ {1} ^ {2}} _ {2} +\ delta {\ mu} _ {3}\ mu} _ {\ mu} _ {3}} ^ {2} _ {3} ^ {2} _ {3} ^ {2} _ {2} -\ delta {2} -\ delta {2} -\ delta {2}\ mu} _ {1}\ delta {\ mu} _ {2} -\ delta {\ mu} _ {1}\ delta {\ mu} _ {3} -\ delta {\ mu} _ {2}\ delta {\ mu} _ {3}} -\ delta {\ mu} _ {3}}\end{split}\]
2.4. Internal variables of the model
The internal variables of the model are nine:
V1 is the \({\mathrm{\epsilon }}_{v}^{p}\) volume plastic deformation;
V2 is the equivalent plastic deformation (deviatoric) \({\stackrel{~}{\mathrm{\epsilon }}}^{p}=\Vert {\stackrel{~}{\mathrm{\epsilon }}}^{p}\Vert\);
V3 is the plasticity indicator;
V4 to V9 are the six components of the \({\mathrm{\epsilon }}^{p}\) plastic deformation tensor.