7. Numerical formulation

We will only deal with the law of viscoplastic behavior with isotropic work hardening. The same thing is obtained for the case of kinematic work hardening, \({R}_{0}\) is then replaced by the kinematic work hardening coefficient \({H}_{0}\).

7.1. Discretization

Knowing the fields \(\sigma\), \(u\) and \(p\) at time \(t\), an implicit scheme is chosen to discretize the equations of the continuous problem in time, except for the work hardening parameters. We note that with implicit discretization, only two points differentiate between the two types of visco-plastic and plastic behavior independent of time:

• The form of the load function, for which we have a complementary term in the case of viscosity (see 6.4.1.1 and 6.4.1.2);

• The presence of the term work-hardening restoration in the evolution of the work-hardening variable for the viscoplastic case.

Moreover, classical incremental plasticity appears to be the borderline case (without associated numerical difficulty) of incremental viscoplasticity when \(\eta ,C\to 0\) and \({\sigma }_{c}\to {\sigma }_{y}\). This type of treatment has already been done by Lorentz [R5.03.05]. To calculate the tangent operators, we will adopt the convention of writing symmetric tensors of order two in the form of vectors with six components. So for a \(a\) tensor:

(7.1)\[ a\ mathrm {=} {} {} ^ {t}\ text {}\ text {}\ left [\ begin {array} {ccc} {a} _ {\ text {yy}}} & {\ text {yy}}} & {a}} & {a}}\ a} _ {\ text {zz}}}\ end {array} {zz}}}\ end {array} {ccc}\ sqrt {2} {a}} _ {\ text {yy}}} & {a} _ {a} _ {a} _ {a} _ {\ text {yy}} & {a}} & {a}} & {a}} & {a}} & {a}} & {a}} & {a}} & {a}} & {a}} & {a}}} &\ sqrt {2} {a} _ {\ text {xz}}} &\ sqrt {2} {a} _ {\ text {yz}}}\ end {array}\ right]\]

7.2. Integration of metallurgical relationships

7.2.1. Expression of constraints

We give the expression for \(\sigma\) according to:

• \(\Delta \varepsilon\) unknown to the issue;

• Known terms such as variables calculated in the previous step (constraints, internal variables, etc.), material characteristics, control variables (temperature, metallurgical phases);

We recall the partition of deviatory deformations:

(7.2)\[ \ tilde {\ varepsilon}\ mathrm {=} {\ tilde {\ varepsilon}} ^ {e} + {\ tilde {\ varepsilon}}} ^ {\ mathit {VP}}} + {\ tilde {\ varepsilon}} ^ {\ mathit {pt}}} ^ {\ mathit {pt}}\]

We separate the stress tensor between its deviatory components and its spherical component:

\[\]

: label: eq-87

sigmamathrm {=}tilde {sigma} +mathit {tr} (sigma)mathit {Id}

Applying Hooke’s law:

\[\]

: label: eq-88

sigmamathrm {=} A {varepsilon} ^ {e}

With its deviatory part:

(7.3)\[ \ tilde {\ sigma}\ mathrm {=} 2\ mu {\ tilde {\ varepsilon}} ^ {e}\]

And its spherical part:

(7.4)\[ \ mathit {tr} (\ sigma)\ mathrm {=}\ mathrm {3K}\ mathit {tr} ({\ varepsilon} ^ {e})\]

By implicit discretization, we obtain:

\[\]

: label: eq-91

tilde {sigma}mathrm {=}}frac {mu} {mu} {{mu} ^ {mathrm {-}}} {tilde {sigma}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathit {VP}}mathrm {-}Delta {tilde {varepsilon}} ^ {mathit {pt}})

By injecting () and ():

\[\]

: label: eq-92

tilde {sigma}mathrm {=}}frac {mu} {mu} {{mu} ^ {mathrm {-}}} {tilde {sigma}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} {tilde {sigma}} {{sigma}} _ {mathit {eq}}}mathrm {-}frac {3} {2}tilde {sigma} {g} {g}} _ {Z}} _ {Z}} ^ {mathit {pt}})

For the spherical part of the stress tensor, we have:

(7.5)\[ \ mathit {tr} (\ sigma)\ mathrm {=}\ frac {\ mathrm {3K}} {{\ mathrm {3K}} ^ {\ mathrm {-}}}}\ mathit {tr}}}\ mathit {tr}} ({\ tr}} ({\ sigma} ({\ sigma} ^ {\ mathrm {-}}}) +\ mathrm {3K}\ mathit {-}}}}}\ mathit {tr}}}}\ mathit {tr}}}}\ mathit {tr}}}}\ mathit {tr} (\ delta\ varete} (\ delta\ varee) psilon)\]

Because plastic deformations are purely deviatory and therefore \(\mathit{tr}({\Delta \varepsilon }^{\mathit{vp}})\mathrm{=}\mathit{tr}({\Delta \varepsilon }^{\mathit{pt}})\mathrm{=}0\). By reordering ():

(7.6)\[ \ tilde {\ sigma}\ mathrm {=}\ frac {1} {1+3\ mu {g} _ {Z} ^ {\ mathit {pt}}}} (\ frac {\ mu} {{\ mu} {{\ mu}} ^ {\ mathrm {\ mu}} ^ {\ mathrm {-}}} +2\ mu\ delta\ tilde {-}} +2\ mu\ delta\ tilde {-}} +2\ mu\ delta\ tilde {\ mu}} {\ mu\ delta\ tilde {-}} +2\ mu\ delta\ tilde {\ mu}} varepsilon}\ mathrm {-} 3\ mu\ mu\ delta p\ frac {\ tilde {\ sigma}} {{\ sigma}} _ {\ mathit {eq}}}})\]

We need to develop the expression of \(\frac{\tilde{\sigma }}{{\sigma }_{\mathit{eq}}}\). We start from the expression for the partition of deviatory deformations, i.e.:

(7.7)\[ \ Delta\ tilde {\ varepsilon}\ mathrm {=}\ Delta {\ tilde {\ varepsilon}} ^ {e} +\ Delta {\ tilde {\ varepsilon}}} ^ {\ mathit {psilon}}} ^ {\ mathit {VP}} +\ Delta {\ tilde {\ varepsilon}} +\ Delta {\ tilde {\ varepsilon}} +\ Delta {\ tilde {\ varepsilon}} ^ {\ mathit {pt}}} ^ {\ mathit {pt}}\]

By injecting the values of the deformation increments:

(7.8)\[ \ Delta\ stackrel {b>} {\ epsilon} =\ left (\ frac {\ stackrel {\ stackrel {}} {\ sigma}} {2\ mu} -\ frac {{\ stackrel {} {} {\ sigma} {\ sigma}}} {\ sigma}}} =\ left) +\ frac {3} {2} {2}\ Delta p\ frac {\ frac {\ frac {2} {2}\ Delta p\ frac {\ frac {\ stackrel {} {\ sigma}} {{\ sigma}} {{\ sigma}} _ {\ mathit {eq}}}} +\ frac {3} {\ sigma} {\ sigma} {\ sigma} {g}} _ {g}} _ {Z}} ^ {\ mathit {pt}}\]

Multiplying by \(2\mu\):

(7.9)\[ 2\ mu\ Delta\ stackrel {} {\ sigma}} {\ epsilon} +\ frac {\ epsilon} +\ frac {2\ mu} ^ {-}} {\ sigma}} ^ {-}} ^ {-} =\ frac {\ epsilon} +\ frac {\ epsilon} +\ frac {2\ mu} {\ mu} {\ mu}} {\ stackrel {-}} {\ sigma}} {\ sigma}} {\ sigma}} {\ mathit {eq}}} {\ sigma}}} {\ mathit {eq}}}} {\ sigma}}}\ left [\ left (1+ +3\ mu {g} _ {Z} ^ {\ mathit {pt}}}\ right) {\ sigma} _ {\ mathit {eq}} +3\ mu\ delta p\ right]\]

The deviatoric elastic stress is set:

(7.10)\[ {\ tilde {\ sigma}} ^ {e}\ mathrm {=} 2\ mu\ delta\ tilde {\ varepsilon} +\ frac {2\ mu} {{2\ mu} {{2\ mu}} ^ {\ mathrm {-}} ^ {\ mathrm {-}} ^ {\ mathrm {-}} ^ {\ mathrm {-}}}\]

And the equivalent elastic stress in the Von Mises sense:

(7.11)\[ {\ sigma} _ {\ mathit {eq}}} ^ {e}} ^ {e}\ mathrm {=}\ left [(1+3\ mu {g}} _ {Z} ^ {\ mathit {pt}}}) {\ mathit {pt}}}) {\ mathit {pt}}}) {\ mathit {pt}}}) {\ sigma}} _ {\ mathit {pt}}) {\ sigma}} _ {\ mathit {pt}}) {\ sigma} _ {\ sigma} _ {\ mathit {pt}}}) {\ sigma} _ {\ sigma}\]

The expression for \(\frac{\tilde{\sigma }}{{\sigma }_{\mathit{eq}}}\) is then equivalent to:

\[\]

: label: eq-100

frac {tilde {sigma}} {{sigma}} {{sigma}} _ {mathit {eq}}}}mathrm {=}frac {{tilde {sigma}}} ^ {e}} ^ {e}}} ^ {e}}} {{e}}} {{sigma}} {{e}}

7.2.2. Expression of plastic increment

The quantities () and () are all known at the current moment, except for increment \(\Delta p\). To find this value, we use the consistency condition:

\[\]

: label: eq-101

mathrm {{}begin {array} {c}overline {f}overline {f}mathrm {=} {sigma} _ {mathit {eq}}mathrm {-}overline {R}overline {R}}\ overline {R}}\ overline {R}}\ overline {R}}overline {R}}mathrm {R}}mathrm {R}\ overline {R}}mathrm {R}}mathrm {R}}mathrm {R}}mathrm {-}mathrm {R}}mathrm {R}\ overline {R}}mathrm {R}}mathrm {Rmathrm {=} {sigma} _ {mathit {eq}} {mathit {eq}}mathrm {-} {overline {sigma}} _ {c}mathrm {-}} {mathrm {-}} {mathrm {-}} {overline {-}} {overline {-}} {overline {sigma}} _ {v}delta pmathrm {=}}} _ {c}mathrm {-}}mathrm {-}} {mathrm {-}} {mathrm {-}} {mathrm {-}} {mathrm {-}} {mathrm {-}} {mathrm {-}} {f} <0\ Delta pmathrm {ge} 0text {if}overline {f}mathrm {=} 0text {with} {overline {f}}}} ^ {overline {f}}} ^ {overline {f}}}} ^ {text {f}}} ^ {text {f}}} ^ {text {f}}} ^ {text {f}}} ^ {text {f}}} ^ {text {f}}} ^ {text {f}}} ^ {text {f}}} ^ {

So all you have to do is express \({\overline{f}}^{\text{*}}\mathrm{=}0\) to find the expression for \(\Delta p\). Only \({\sigma }_{\mathit{eq}}\), \(\overline{R}\), and the viscous stress \({\overline{\sigma }}_{v}\) depend on \(\Delta p\). Or in incremental form:

\[\]

: label: eq-102

{overline {f}} ^ {text {*}}}mathrm {=} {sigma} _ {mathit {eq}}mathrm {-}overline {R} ({p} ({p}} ^ {p} ^ {mathrm {-}} ^ {mathrm {-}}} +mathrm {-}} +mathrm {-}} +mathrm {-}} +mathrm {-}} +mathrm {-}} +mathrm {-}}mathrm -}overline {eta} {(frac {Delta p} {Delta p} {Delta t})} ^ {frac {1} {overline {n}}}}}mathrm {Delta p} {Delta p} {Delta t})} ^ {frac {1} {overline {n}}}}}mathrm {=} 0

Using the expression (), we find:

\[\]

: label: eq-103

{sigma} _ {mathit {eq}}}mathrm {=}frac {{sigma} _ {mathit {eq}} ^ {e}mathrm {-} 3muDelta p} 3muDelta p} {1+3muDelta p}} {1+3mudelta p}} {1+3mu {g} _ {Z}} ^ {mathit {pt}}}

So we need to find the zero of the following function to determine \(\Delta p\):

\[\]

: label: eq-104

{overline {f}} ^ {text {*}}}mathrm {*}}mathrm {=}}frac {{sigma}} ^ {e}mathrm {-} 3muDelta p} 3muDelta p}} {3muDelta p}} {1+3mu {g} _ {Z} _ {Z} ^ {mathit {pt}}}mathrm {-}}mathrm {-} 3muDelta p} {3muDelta p} {1+3muDelta p} {1+3mu {g} _ {Z} _ {Z} ^ {mathit {pt}}}}mathrm {-}}mathrm {-}} 3muDelta p ({p} ^ {mathrm {-}})mathrm {-}}mathrm {-}overline {-} {overline {sigma}} _ {c}mathrm {-}mathrm {-}mathrm {-}overline {-}overline {-}}overline {eta} {eta} (frac {Delta p} {delta t})} ^ {frac {frac {1} {frac {1} {-}overline {-}}mathrm {-}}mathrm {-}}mathrm {-}}mathrm {-}overline {-}overline {-}overline {eta}} {line {n}}}mathrm {=} 0

This equation is non-linear in \(\Delta p\). The resolution is done in code_aster by a secant method with search interval [R5.03.05].

7.3. Quantities in prediction

The purpose of this paragraph is to calculate the quantities needed in prediction. Among other things, the tangent operator \({K}_{i-1}\) (calculation option RIGI_MECA_TANG called at the first iteration of a new load increment) is evaluated based on the results known at the previous instant \({t}_{i\mathrm{-}1}\):

\[\]

: label: eq-105

{K} _ {imathrm {-} 1}mathrm {=} {frac {dsigma} {dvarepsilon}mathrm {mid}}} _ {{t}} _ {{t} _ {imathrm {-} 1}}

As we make the quasi-static hypothesis, the different quantities depend on time only in an implicit way, via the dependence of the material parameters on the control variables [1] _ \(\beta (t)\), which are themselves functions of time. In the present situation, there are two control variables: temperature \(T\) and metallurgical phases \(Z\). For \({n}_{\mathrm{varc}}\) command variables, we write the total differential of a quantity \(a\):

\[\]

: label: eq-106

frac {da} {dt}mathrm {=}frac {mathrm {partial} a} {mathrm {partial} t} +mathrm {sum} _ {jmathrm {=}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}mathrm {=}}mathrm {=}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {=}}mathrm {delta {beta} ^ {j}}mathrm {cdot}frac {delta {beta} ^ {j}}} {delta t}

We start from the expression of Hooke’s law incremental, for its deviatory part:

\[\]

: label: eq-107

dot {tilde {sigma}}mathrm {=}} 2mu (dot {tilde {varepsilon}}mathrm {-} {dot {tilde {varepsilon}}}}}mathrm {-}}mathrm {-}} {dot {tilde {varepsilon}}} {tilde {varepsilon}}}} ^ {tilde {varepsilon}}}} ^ {tilde {varepsilon}}}} ^ {tilde {varepsilon}}}} ^ {tilde {varepsilon}}}} ^ {tilde {varepsilon}}} {mathit {pt}})

The main difficulty with this expression is that viscoplastic deformation \({\dot{\tilde{\varepsilon }}}^{\mathit{vp}}\) has as its unknown the plastic multiplier, which is itself an unknown of the problem. For the spherical part, on the other hand, the expression is trivial because it only depends on thermal deformation and elastic deformation:

\[\]

: label: eq-108

mathit {tr} (dot {sigma})mathrm {=}mathrm {=}mathrm {3K}mathit {tr} (dot {varepsilon})mathrm {-}mathrm {-}mathrm {3K}}mathrm {3K}}mathit {tr} ({dot {varepsilon}})mathrm {-}}mathrm {3K}}mathrm {3K}}mathit {tr} ({dot {varepsilon}}) ^ {mathit {th}})

With the expression for thermal deformation given by ().

7.3.1. Plastic multiplier

To establish the expression for the plastic multiplier, we write the consistency condition \(\dot{\overline{f}}\mathrm{=}0\):

\[\]

: label: eq-109

dot {overline {f}}mathrm {=}frac {d} {mathit {dt}} ({sigma} _ {mathit {eq}}}mathrm {-}mathrm {-}overline {-}overline {sigma} _ {eq}}}}mathrm {-}})mathrm {-}}overline {R}}overline {R}}overline {R}}mathrm {R}}overline {R}}mathrm {R}}overline {R}}mathrm {R}}overline {R}}mathrm {R}}overline {

The function \(\overline{f}\) depends on the equivalent Von Mises constraint, which is purely deviatory and has the following values:

\[\]

: label: eq-110

{sigma} _ {mathit {eq}}mathrm {=}} {(frac {3} {2}tilde {sigma}mathrm {:}tilde {sigma})}tilde {sigma})}}} ^ {1mathrm {/} 2}

The contribution is then written:

\[\]

: label: eq-111

frac {{dsigma} _ {mathit {eq}}}} {dt}} {dt}mathrm {=}frac {{mathrm {partial}sigma}} _ {mathit {eq}}}}} {mathit {eq}}}}} {mathrm {partial}}} {partial} t}} {partial} t} +frac {{deltasigma} _ {mathit {eq}}} {mathit {eq}}}} {mathrm {partial}}}} {partial}}mathrm {cdot}frac {delta T} {delta T} {delta T} +frac {{deltasigma}} _ {mathit {eq}}} {delta Z}} {delta Z}mathrm {delta T}mathrm {delta T}mathrm {delta T}mathrm {delta T} {delta T}

We make the assumption (usual in code_aster) that the variation in the Hooke \(A\) matrix as a function of control variables is negligible. That is to say that:

\[\]

: label: eq-112

frac {{deltasigma} _ {mathit {eq}}}} {delta T}mathrm {=}frac {{deltasigma}} _ {mathit {eq}}}} {mathit {eq}}}} {delta Z}mathrm {=} 0

In the end, the \({\dot{\sigma }}_{\mathit{eq}}\) contribution is then worth:

(7.12)\[ \ frac {{d\ sigma} _ {\ mathit {eq}}}} {dt}} {dt}\ mathrm {=} {\ dot {\ sigma}} _ {\ mathit {eq}}\ mathrm {=}\ mathrm {=}}\ frac {3} {2}\ frac {\ tilde {3} {2}}\ frac {\ tilde {\ sigma}}\ mathrm {:} A {\ dot {\ tilde {\ tilde {\ tilde {\ varepareparepareparepareparepave} Silon}}} ^ {e}} {{\ sigma}} _ {\ mathit {eq}}}\]

We are now dealing with the case of the \(\overline{R}\) work hardening term. We are going to overlook the variation in the work hardening module in relation to the control variables. That is to say that:

\[\]

: label: eq-114

frac {deltaoverline {R}} {delta T} {delta T}mathrm {=}frac {deltaoverline {R}} {delta Z}}mathrm {=} 0

The total time difference is therefore written as:

\[\]

: label: eq-115

frac {doverline {R}} {mathit {dt}} {mathit {dt}}mathrm {=}} _ {kmathrm {=} 1} ^ {5} {dot {Z}} {Z}}}} _ {k}}} _ {r} _ {k} +mathrm {sum}} _ {k} +mathrm {sum} _ {k} _ {k} _ {kmathrm {=} 1} ^ {5} {dot {r}}} _ {k} {R} _ {mathrm {0,} k} {Z} _ {k}

The variation of metallurgical phases is overlooked. The second term involves using the law of multiphase work hardening written in () without the term viscous restoration, which we will write in condensed form:

\[\]

: label: eq-116

{dot {r}} _ {k}mathrm {=} {dot {p}} _ {k} + {g} _ {k}} ^ {text {re}, m} ^ {text {re}, m}

So we have:

\[\]

: label: eq-117

frac {doverline {R}} {mathit {dt}} {mathit {dt}}mathrm {=}} _ {kmathrm {=} 1} ^ {5} {dot {Z}} {Z}}}} _ {k}}} _ {r} _ {k} +mathrm {sum}} _ {k} +mathrm {sum} _ {k} _ {k} _ {kmathrm {=} 1} ^ {5} (dot {p} + {g} + {g} _ {k} ^ {text {re}, m}) {R} _ {mathrm {0,} {0,} k} {Z} _ {k}

We recognize the equivalent module \({\overline{R}}_{0}\):

(7.13)\[ {\ overline {R}} _ {0}\ mathrm {=}\ mathrm {=}\ mathrm {\ sum} _ {k\ mathrm {=} 1} ^ {5} {R} _ {R} _ {R} _ {R} _ {R} _ {R} _ {R} _ {R} _ {R} _\]

The other terms will be grouped together in function \(B\):

(7.14)\[ \ frac {d\ overline {R}} {\ mathit {dt}} {\ mathit {dt}}\ mathrm {=} {\ overline {R}}} _ {0}\ dot {p} +B\]

Since the effects of the variation of metallurgical phases are overlooked, this term \(B\) is assumed to be zero. Finally:

(7.15)\[ \ frac {d\ overline {R}} {\ mathit {dt}}\ mathrm {=}\ dot {\ overline {R}}\ mathrm {=} {\ overline {R}} {\ overline {R}}} _ {0}\ dot {p}\]

Finally, we consider the case of the elastic limit \({\sigma }_{y}\). We are going to make a final simplification by assuming that this term does not vary as a function of metallurgical structure and temperature. That is to say:

\[\]

: label: eq-121

frac {{doverline {sigma}}} _ {y}}} {mathit {dt}} {mathit {dt}}mathrm {sigma}} _ {y}mathrm {=}frac {{mathrm {=}}frac {mathrm {sigma}}dot {t}}dot {t} +frac {{mathrm {partial}overline {sigma}} _ {sigma}} _ {y}} {mathrm {partial} T}dot {T} +frac {{mathrm {partial}partial}overline {partial}}overline {sigma}}} _ {y}}} {mathrm {partial} Z}dot {Z}mathrm {partial}mathrm {partial}mathrm {partial}}mathrm {partial}}mathrm {partial}}mathrm {=} C

Quantity \({\overline{\sigma }}_{y}\) is a time constant. Its partial derivative with respect to time is therefore zero. The other quantities (dependence on temperature and metallurgical phases) are grouped together in function \(C\). This term \(C\) will be assumed to be void. In the end, we will have trivially:

\[\]

: label: eq-122

frac {{doverline {sigma}}} _ {y}} {mathit {dt}}mathrm {=} {dot {sigma}}} _ {y}mathrm {=} 0

By taking () and injecting the results of the differentiation of the terms (), () and () into it, we finally have:

(7.16)\[ \ dot {\ overline {f}} =\ frac {3} {2}\ frac {\ stackrel {2}\ frac {\ stackrel {2}} =\ frac {3} {\ epsilon}} {\ epsilon}}}\ frac {\ stackrel {2}}\ frac {\ stackrel {2}}}\ frac {\ stackrel {2}}}\ frac {\ stackrel {2}}} {\ epsilon}}}}} {\ epsilon}}}}} ^ {e}}}} {{e}}} {\ sigma} _ {0}} - {\ overline {R}}} _ {0}\ dot {p} =0\]

The deviatoric component of Hooke’s law is written in incremental form:

(7.17)\[ \ dot {\ tilde {\ sigma}}\ mathrm {=}} 2\ mu {\ dot {\ tilde {\ varepsilon}}}}} ^ {e}\]

This gives a new (deviatoric) expression for the consistency condition:

\[\]

: label: eq-125

dot {overline {f}} =3mufrac {stackrel {▲} {sigma}mathrm {:}left (dot {stackrel {} {epsilon}} {epsilon}}} - {dot {epsilon}}} - {dot {epsilon}}} - {dot {stackrel}} - {dot {stackrel}} - {dot {stackrel}} - {dot {stackrel}}} {epsilon}}}} ^ {mathit {pt}}}right)} {{sigma} _ {mathit {eq}}}} - {overline {R}}} _ {0} _ {0}dot {p} =0

The elastic deformation tensor must be expressed using the expression ():

(7.18)\[ {\ dot {\ tilde {\ varepsilon}}}} ^ {e}} ^ {e}\ mathrm {=}\ dot {\ tilde {\ varepsilon}}\ mathrm {-} {\ dot {\ dot {\ tilde {\ varepsilon}}}} ^ {\ tilde {\ varepsilon}}} ^ {\ mathit {VP}}\ mathrm {-} {\ dot {\ tilde {\ varepsilon}}} {\ dot {\ tilde {\ varepsilon}}} {\ dot {\ tilde {\ varepsilon}}}} ^ {\ mathit {VP}}\ mathrm {-} {\ dot {\ tilde {\ varepsilon} Silon}}} ^ {\ mathit {pt}}\]

The first term for plastic deformation is developed:

\[\]

: label: eq-127

3mufrac {tilde {sigma}mathrm {:} {dot {tilde {varepsilon}}}} ^ {mathit {VP}}} {{sigma}} _ {sigma} _ {mathit {eq}}mathrm {}}}}mathrm {=}}mathrm {=} 3mufrac {tilde {sigma}}} {{sigma}} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} {3} {2}dot {p}frac {tilde {sigma}} {{sigma}} _ {mathit {eq}}}}} {{sigma}} _ {sigma} _ {mathit {eq}}}

Using ():

(7.19)\[ 3\ mu\ frac {\ tilde {\ sigma}\ mathrm {:} {\ dot {\ tilde {\ varepsilon}}}} ^ {\ mathit {VP}}} {{\ sigma}}} {{\ sigma}} _ {\ sigma} _ {\ mathit {eq}}}} {\ mathit {eq}}}} {\ mathit {eq}}}}\ mathrm {=} 3\ mu\ dot {p}}\]

We develop the second term for transformation plasticity ():

\[\]

: label: eq-129

3mufrac {tilde {sigma}mathrm {:} {dot {tilde {varepsilon}}}} ^ {mathit {pt}}} {{sigma}} _ {sigma} _ {mathit {eq}}mathrm {}}}}mathrm {=}}mathrm {=} 3mufrac {tilde {sigma}}} {{sigma}} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} {3} {2}tilde {sigma} {g} {g} _ {Z} _ {Z} ^ {mathit {pt}} (Z,dot {Z})}} {{sigma}} _ {sigma} _ {mathit {eq}}}

It is important to note that the term \({g}_{Z}^{\mathit{pt}}\) depends on the increment of the metallurgical phases \(\dot{Z}\). Afterwards, to simplify the writing, we will not write it explicitly. Always using ():

\[\]

: label: eq-130

3mufrac {tilde {sigma}mathrm {:} {dot {tilde {varepsilon}}}} ^ {mathit {pt}}} {{sigma}} _ {sigma} _ {sigma} _ {g}} _ {Z} _ {Z}}} {mathit {eq}} _ {Z}} ^ {mathit {pt}}

Finally:

\[\]

: label: eq-131

dot {overline {f}}mathrm {=}} 3mufrac {tilde {sigma}mathrm {:}dot {tilde {varepsilon}}}} {{sigma}}} {{sigma}}} {sigma}}} {mathit {eq}} _ {mathit {eq}}} {g} _ {Z} ^ {mathit {pt}}}mathrm {-} ({overline {R}} _ {0} +3mu)dot {p}mathrm {=} 0

This allows us to express \(\dot{p}\):

\[\]

: label: eq-132

dot {p} =frac {3mu} {{overline {R}} {{overline {R}}} _ {0} +3mu}left (frac {stackrel {} {sigma} {sigma} {sigma} {sigma} {sigma} {sigma} {sigma}} {sigma}} {sigma}} {sigma}}mathrm {::}}mathrm {::}:}dot {overline {R}}} {overline {R}}} {stackrel {R}}} {sigma}}mathrm {::}:}mathrm {:}:}dot {overline {R}}} {overline {R}} _ {mathit {eq}} {g} _ {Z} ^ {mathit {pt}}right)

This expression is different from the one given by () because it is an approximation. In fact, the consistency condition was considered without the term viscous as is customary to proceed. But this expression does not need to be exact since it is the tangent matrix in prediction and has the advantage of being an analytical formula, unlike () which requires the use of an algorithm for finding zero on a non-linear function.

7.3.2. Stress increment

From the definition of the visco-plastic deformation increment \({\dot{\tilde{\varepsilon }}}^{\mathit{vp}}\):

\[\]

: label: eq-133

{dot {tilde {varepsilon}}}} ^ {mathit {VP}}mathrm {=}mathrm {{}begin {array} {ccc}frac {3} {3} {2}frac {mathit {eq} {2}}dot {p}}dot {p}}frac {tilde {sigma}} {sigma} _ {mathit {eq}}} {mathit {eq}}}frac {tilde {sigma}} _ {mathit {eq}}} {mathit {eq}}}frac {tilde {eq}}} &text {si} & {sigma} _ {mathit {eq}}}mathrm {-}mathrm {-}mathrm {-} {overline {sigma}} _ {c}mathrm {=} =} 0\ =} 0\ 0&0&text {-}text {-} 0\ 0&text {if} & {sigma}} _ {mathit {eq}}mathrm {-}mathrm {-}mathrm {-}overline {R}}mathrm {-} {overline {sigma}} _ {c} <0end {array}

And from the expression for \(\dot{p}\) obtained previously, we get:

(7.20)\[\begin{split} {\ dot {\ varepsilon}}} ^ {\ mathit {VP}}\ mathrm {=}\ mathrm {\ {}\ begin {array} {ccc}\ frac {9\ mu} {2 ({\ overline {R}} {2 ({\ overline {R}}}} _ {0} +3\ mu)}\ langle\ frac {\ tilde {\ sigma}\ mathrm {:}}\ mathrm {:}\ dot {\ tilde {\ varepsilon}}}} {{\ sigma}}} {{\ sigma}}} {\ mathit {eq}}} {\ mathit {eq}} {g} _ {Z}} ^ {\ Z} ^ {\ Z}} ^ {\ mathit {pt}} {\ mathit {pt}}}\ mathrm {-} {\ sigma}} _ {\ mathit {eq}} {g} _ {Z}} _ {Z}} ^ {\ Z} ^ {\ mathit {pt}} ^ {\ mathit {pt}}}\ rangle\ frac {\ tilde {\ sigma}} {{\ sigma}} {{\ sigma}} {\ sigma}} {\ sigma}} {\ sigma}} {\ sigma} q}}} &\ text {si} & {\ sigma}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ overline {\ sigma}} _ {c}\ mathrm {=}} 0\\ mathit {eq}}} _ {c}\ mathrm {=}} 0\\ 0&\ text {-}\ sigma} _ {\ mathit {eq}}}\ mathrm {=} 0\\ 0&\ text {si}} & {\ sigma} _ {\ mathit {eq}}}\ mathrm {-}\ overline {R}\ mathrm {-} {\ overline {\ sigma}} _ {c} <0\ end {array}\end{split}\]

Thanks to the expression of incremental Hooke’s law:

(7.21)\[ \ dot {\ tilde {\ sigma}}\ mathrm {=}} 2\ mu (\ dot {\ tilde {\ varepsilon}}\ mathrm {-} {\ dot {\ tilde {\ varepsilon}}}}}\ mathrm {-}}\ mathrm {-}} {\ dot {\ tilde {\ varepsilon}}} {\ tilde {\ varepsilon}}}} ^ {\ tilde {\ varepsilon}}}} ^ {\ tilde {\ varepsilon}}}} ^ {\ tilde {\ varepsilon}}}} ^ {\ tilde {\ varepsilon}}}} ^ {\ tilde {\ varepsilon}}} {\ mathit {pt}})\]

By injecting the expression for plastic deformation () and for transformation plasticity, we obtain the expression for the stress increment:

(7.22)\[ \ dot {\ stackrel {459} {\ sigma}} {\ sigma}} =2\ mu\ sigma}} =2\ mu\ sigma}} =2 {\ sigma} _ {\ sigma} _ {\ sigma} _ {\ mathit {eq}}} =2\ mu\ sigma}} =2\ mu\ sigma}} =2\ mu\ sigma}} = 2\ mu\ sigma}} = 2\ mu\ sigma}} _ {\ stackrel {\ epsilon}} -\ frac {\ sigma} {2 {\ sigma} _ {\ sigma} _ {\ sigma} _ {\ sigma} {2\ sigma} _ {\ sigma} _ {\ sigma} {2\ sigma} _ {\ sigma} _ {\ sigma} {2\ sigma} _ {\ sigma} _ {\ sigma} {ackrel {459} {\ sigma}\ mathrm {:}\ dot {\ stackrel {:}\ dot {\ stackrel {}}} {\ sigma} _ {\ mathit {eq}}}} - {\ sigma}} _ {\ mathit {eq}} _ {g}} _ {\ epsilon}}} {\ sigma}} {\ sigma}}\ rangle\ stackrel {}}} - {\ sigma}} - {\ sigma}} - {\ sigma}} - {\ sigma}} - {\ sigma} _ {\ sigma}} - {\ sigma}} - {\ sigma}} - {\ sigma}} - {\ sigma} _ {\ sigma}} - {\ sigma}} - {\ sigma}} - {\ sigma}} - {\ sigma -\ frac {3} {2} {g} {g} _ {Z} _ {Z} ^ {\ mathit {pt}}\ stackrel {} {\ sigma}\ right]\]

The expression for \(\dot{\tilde{\sigma }}\) depends on the sign of the term (charge-discharge criterion) following:

(7.23)\[ \ frac {\ tilde {\ sigma}\ mathrm {:}\ dot {\ tilde {\ varepsilon}}} {{\ sigma} _ {\ mathit {eq}}}}\ mathrm {-} {-} {\ sigma}} _ {\ mathit {eq}}} {g} _ {Z} ^ {\ mathit {eq}}} {\ mathit {eq}}} {\ mathit {pt}}}\]

We can simplify by saying that the discharge or the charge do not depend on the plasticity of transformation, so we will approximate:

\[\]

: label: eq-138

langlefrac {tilde {sigma}mathrm {:}dot {tilde {varepsilon}}} {sigma} _ {mathit {eq}}}}mathrm {-} {sigma}} _ {mathit {eq}} _ {Z} ^ {mathit {pt}}}}mathrm {-} {sigma}}}mathrm {-} {sigma}} _ {mathit {pt}}}mathrm {-} {sigma}} _ {mathit {pt}}}mathrm {-} {sigma}} _ {mathit {pt}}}mathrm {-} {sigma}} _ {mathit {pt}}} thrm {approx}frac {langletilde {sigma}mathrm {:}dot {tilde {varepsilon}}rangle} {{sigma} _ {mathit {eq}} _ {mathit {eq}} _ {sigma} _ {sigma} _ {Z} _ {Z}} _ {Z}} {mathit {eq}} _ {Z}} _ {mathit {eq}} _ {Z}} {mathit {eq}} _ {Z}} {mathit {eq}} _ {Z}} {mathit {eq}} _ {Z}} {mathit {eq}} _ {Z}} {mathit {eq}} _ {{pt}}

In landfill, the second term will be overlooked and in charge, it will be added to the other terms of ():

(7.24)\[\begin{split} \ dot {\ stackrel {} {\ sigma}}} =\ {\ begin {sigma}}} =\ {\ begin {array} {ccc} 2\ mu\ left [\ dot {\ stackrel {}} {\ epsilon}}} -\ frac {9\ mu} {\ sigma}} {\ epsilon}}} -\ frac {9\ mu} {\ sigma}} -\ frac {9\ mu} {\ sigma}} -\ frac {9\ mu} {\ sigma}} -\ frac {9\ mu} {\ sigma}} -\ frac {9\ mu} {\ sigma}} {\ epsilon}} -\ frac {9\ mu} {\ sigma}} {\ epsilon}} -\ frac {9\ mu 459} {\ epsilon}}\ rangle} {\ left ({\ overline {R}}} _ {0} +3\ mu\ right) {\ sigma} _ {\ mathit {eq}} ^ {2}} ^ {2}}}\ stackrel {2}}}\ stackrel {229}} {\ sigma} -\ frac {3} {2} {g} _ {Z}} ^ {\ mathit {pt}}}\ stackrel {} {\ sigma}\ right] &\ text {in landfill} &\\ 2\ mu\ left [\ dot {\ stackrel {} {\ epsilon}} -\ frac {9\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ langle\ stackrel {} {\ epsilon}}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {\ mu} {2}\ frac {}}\ rangle} {\ left ({\ overline {R}}} _ {0} +3\ mu\ right) {\ sigma} _ {\ mathit {eq}}} ^ {2}}}\ stackrel {} {\ sigma} -\ frac {3 {\ overline {R}}} _ {0}} {2\ left ({\ overline {R}}} _ {0} +3\ mu\ right)} {g} _ {Z}}}\ frac {3 {\ overline {R}}}\ frac {\ overline {R}}} {\ sigma}} _ {0} +3\ mu\ right)} {g} _ {\ right)} {g} _ {Z}}} {Z} ^ {\ Z} ^ {\ mathit {pt}} ^ {\ mathit {pt}}} {\ mathit {pt}}} {\ sigma}\ right] &\ text {in charge}} {g} _ {\ right]} {g} _ {\ right}\end{split}\]

The parameter \(d\) is introduced, which is equal to 1 if we are plasticizing and if we are in charge at the time \(t\) and 0 in the opposite case:

(7.25)\[ \ dot {\ stackrel {459} {\ sigma}} {\ sigma}} =2\ mu\ left [\ dot {\ stackrel {459} {\ epsilon}} -\ frac {9\ mu} {2}\ frac {\ frac {\ langle\ stackrel {\ sigma}} {\ stackrel} {\ epsilon}}\ frac {\ epsilon}}\ frac {\ epsilon}}\ frac {\ langle\ stackrel {\ sigma}} =2\ mu\ sigma}} =2\ mu\ left [\ dot {\ stackrel {4\ mu} {2} {\ sigma}} {\ sigma}} =2\ mu\ sigma}} =2\ mu\ sigma}} =2\ mu\ left [\ dot {angle} {\ left ({\ overline {R}}} _ {0}} +3\ mu\ right) {\ sigma} _ {\ mathit {eq}} ^ {2}}\ stackrel {} {\ sigma} {\ sigma} -\ frac {3} {sigma}} -\ frac {3} {sigma}} -\ frac {3} {\ sigma} -\ frac {3} {2} {2} {2} {g} {2} {g} _ {Z} _ {Z}} ^ {\ mathit {pt}}}\ left (Z,\ dot {Z}\ sigma}} -\ frac {3} {sigma}} -\ frac {3} {sigma} -\ frac {3} {2} {2} {2} {g} (1-d\ frac {3\ mu} {3\ mu} {3\ mu + {\ mu+ {R}}} _ {0}}\ right)\ stackrel {} {\ sigma}\ right]\]

We note that \(\dot{\tilde{\sigma }}\) is an affine function of \(\dot{\tilde{\varepsilon }}\) but also that there is a part that does not depend on \(\dot{\tilde{\varepsilon }}\) but on the increment of the metallurgical phases \(\dot{Z}\).

7.3.3. Second member in prediction — Contribution of metallurgy

Because of the dependence of () on the increment of metallurgical phases, there is a \(\Delta {L}^{\mathit{pt}}\) contribution to the second member [2] _ of the temporal variation of the stress tensor. On simple test cases for which there is an analytical solution, we found that neglecting the second member \(\Delta {L}^{\mathit{pt}}\) could lead, in order to converge, to a large number of iterations. That is why this term is taken into account for the prediction phase. The contribution comes from transformation plasticity, which is a purely deviatoric quantity. In the end, we get:

\[\]

: label: eq-141

Delta {L} ^ {mathit {pt}}} = {int}} = {int}} _ {pt}}} = {mathit {pt}}left (Z,dot {Z}right)left (Z,dot {Z}right)left)left (1-dint} _ {Z}}right)left (1-dint} _ {Z}}right)left (1-dint} _ {Z}}right) {int} _ {Z}}right) {int} _ {Z}}right) {int} _ {Z}}right) {int} _ {Z}}right) {int} _ {Z}}right) {int} _ {Z}}right) ackrel {} {sigma}right]deltaepsilon domega

With \(\delta \varepsilon\) virtual deformation.

7.3.4. Tangent matrix in prediction (option RIGI_MECA_TANG)

Recall that the tangent operator \({K}_{i-1}\) (calculation option RIGI_MECA_TANG called at the first iteration of a new load increment) is evaluated on the basis of the results known at the previous instant \({t}_{i\mathrm{-}1}\):

\[\]

: label: eq-142

dot {sigma}mathrm {=} {K} _ {imathrm {-} 1}dot {varepsilon}

In charge, we assume that we are plasticizing and therefore we have:

(7.26)\[ \ langle\ tilde {\ sigma}\ mathrm {:}\ dot {\ tilde {\ varepsilon}}\ rangle\ mathrm {=}\ tilde {\ sigma}\ mathrm {:}\ mathrm {:}\ dot {\ tilde {\ varepsilon}}\]

The deviatory part of the prediction matrix is therefore equal to:

\[\]

: label: eq-144

{tilde {K}} _ {imathrm {-} 1}mathrm {-} 1}mathrm {=} 2muleft [Imathrm {-}frac {9mu} {2 ({overline {R}}} {overline {R}}} {overline {R}}} {overline {R}}} {overline {R}}} {overline {R}}} {overline {R}}} {overline {R}}} _ {overline {R}}} _ {overline {R}}} _ {overline {R}}} _ {overline {R}}} _ {overline {R}}} _ {overline {R}}}mathrm {otimes}frac {tilde {sigma}} {{sigma} _ {mathit {eq}}}}right]

The expression () gives the value of \(\frac{\tilde{\sigma }}{{\sigma }_{\mathit{eq}}}\). The spherical part is equal to:

\[\]

: label: eq-145

frac {1} {3}mathit {tr} ({K}} _ {imathrm {-} 1})mathrm {=} KI

Finally:

(7.27)\[ {K} _ {i\ mathrm {-} 1}\ mathrm {-} 1}\ mathrm {=} 2\ mu\ left [I\ mathrm {-}\ frac {9\ mu} {2 ({\ overline {R}}}} _ {\ overline {R}}}} _ {\ overline {R}}} _ {\ overline {R}}} _ {0} +3\ mu)}\ frac {\ tilde {\ sigma}} {\ sigma} _ {\ mathit {eq}}}\ mathrm m {\ otimes}\ frac {\ tilde {\ sigma}} {{\ sigma}} _ {\ mathit {eq}}}}\ right] +KI\]

7.4. Quantities in correction

7.4.1. Tangent matrix in correction (option FULL_MECA)

The tangent operator \({K}^{n}\) (calculation option FULL_MECA called at each Newton’s iteration) is such that:

\[\]

: label: eq-147

{K} ^ {n}mathrm {=}frac {mathrm {partial}sigma} {mathrm {partial}partial}varepsilon}

This matrix has two contributions, deviatoric and spherical:

\[\]

: label: eq-148

{K} ^ {n}mathrm {=}frac {mathrm {partial}sigma} {mathrm {partial}varepsilon}mathrm {=}frac {frac {mathrm {partial}frac {partial}frac {partial}frac {1} {3}frac {mathrm {partial}mathit {tr} (sigma)} {mathrm {partial}varepsilon} I

The spherical part is trivial:

\[\]

: label: eq-149

frac {1} {3}frac {mathrm {partial}mathit {tr} (sigma)} {mathrm {partial}varepsilon} Imathrm {partial}varepsilon} Imathrm {partial}partial}

With K the compressibility module. Now we need to express the deviatoric part of the tangent matrix. We have:

(7.28)\[ \ frac {\ mathrm {\ partial}\ tilde {\ sigma}} {\ mathrm {\ partial}\ varepsilon}\ mathrm {=}\ frac {\ mathrm {\ partial}\ tilde {\ partial}\ tilde {\ partial}\ tilde {\ sigma}} {\ mathrm {\ sigma}} {\ mathrm {\ partial}}\ frac {\ mathrm {\ partial}}\ frac {\ mathrm {\ partial}}\ tilde {\ varepsilon}} {\ mathrm {\ partial}\ varepsilon}\]

We start again with the expression of deviatory constraints:

\[\]

: label: eq-151

tilde {sigma}mathrm {=}} 2mu (tilde {varepsilon}mathrm {-} {tilde {varepsilon}}} ^ {mathit {VP}}}mathrm {-} {tilde {varepsilon}}} ^ {mathit {pt}}} ^ {mathit {pt}})

And we drift:

(7.29)\[ \ frac {\ mathrm {\ partial}\ tilde {\ sigma}} {\ mathrm {\ partial}\ tilde {\ varepsilon}}\ mathrm {=}\ frac {\ mathrm {\ partial}}} {\ mathrm {\ partial}} {\ mathrm {\ partial}}\ tilde {\ varepsilon}}\ left [2\ mu (\ tilde {\ partial}}\ left [2\ mu (\ tilde {\ partial}} arepsilon}\ mathrm {-} {\ tilde {\ varepsilon}}} ^ {\ mathit {VP}}\ mathrm {-} {\ tilde {\ varepsilon}} {\ tilde {\ varepsilon}}}} ^ {\ mathit {pt}})\ right]\]

We find:

\[\]

: label: eq-153

frac {mathrm {partial}tilde {varepsilon}} {mathrm {partial}tilde {varepsilon}}}mathrm {=} I

For transformation plasticity:

\[\]

: label: eq-154

frac {{mathrm {partial}tilde {varepsilon}}} ^ {mathit {pt}}} {mathrm {partial}tilde {varepsilon}}mathrm {=}mathrm {=}}frac {=}}frac {3} {2} {g}} _ {Z} ^ {mathit {pt}}}mathrm {=}}mathrm {=}}frac {3} {2} {g}} _ {Z} ^ {mathit {pt}}}mathrm {=}}mathrm {=}}frac {3} {2} {g}} _ {Z} ^ {mathit {pt}} m {partial}tilde {sigma}}} {mathrm {partial}tilde {varepsilon}}

For visco-plasticity:

\[\]

: label: eq-155

frac {{mathrm {partial}tilde {varepsilon}}} ^ {mathit {VP}}} {mathrm {partial}tilde {varepsilon}}mathrm {=}mathrm {=}frac {3} {2}frac {mathrm {partial}} {mathrm {partial}} {mathrm {partial}} {mathrm {partial}}}mathrm {partial}}mathrm {partial}}mathrm {partial}}mathrm {partial}}mathrm {partial}}mathrm {partial}}mathrm {partial}}mathrm tilde {varepsilon}} (Delta pfrac {tilde {sigma}} {{sigma}} _ {mathit {eq}}})mathrm {=}frac {3} {2}left [frac {frac {sigma}} {mathrm {partial}}tilde {partial}tilde {2}left [frac {sigma}} {mathit {eq}}}})mathrm {partial}}frac {3} {2} {2}left [frac {sigma}} {mathit {eq}}}})mathrm {partial}}frac {3} {2} {2}left [varepsilon}}mathrm {otimes}frac {tilde {sigma}} {{sigma} _ {mathit {eq}}}} +Delta pfrac {mathrm {partial}}frac {partial}} {mathrm {partial}} {mathrm {partial}}tilde {varepsilon}} (frac {tilde {sigma}} {{sigma} _ {mathit {eq}}})right]

We need to evaluate \(\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{\varepsilon }}(\frac{\tilde{\sigma }}{{\sigma }_{\mathit{eq}}})\) but the expression () established the value of \(\frac{\tilde{\sigma }}{{\sigma }_{\mathit{eq}}}\):

\[\]

: label: eq-156

frac {tilde {sigma}} {{sigma}} {{sigma}} _ {mathit {eq}}}mathrm {=}frac {{tilde {sigma}}} ^ {e}} ^ {e}}} ^ {e}}} {{e}}} {{sigma}} {{e}}

By applying the usual derivation rules:

\[\]

: label: eq-157

frac {mathrm {partial}} {mathrm {partial}} {mathrm {partial}}tilde {sigma}} {{sigma} _ {sigma} _ {mathit {eq}}}})mathrm {partial}}} {mathit {eq}}} {mathit {eq}}})mathrm {eq}}})mathrm {eq}}})mathrm {eq}}})mathrm {eq}}})mathrm {eq}}})mathrm {=}frac {mathrm {partial}} de {varepsilon}} (frac {{tilde {sigma}}} ^ {e}} {{sigma} _ {mathit {eq}}} ^ {e}})mathrm {=}}frac {=}frac {1}} {1} {{1}} {{sigma}} _ {mathit {eq}}} ^ {e}})mathrm {=}}frac {1}} {1} {1} {{sigma} _ {mathit {eq}}} ^ {e})} ^ {e}})mathrm {=}}\ frac {1}} {1} {1} {sigma} _ {sigma}} ^ {e}})} _ {mathit {eq}} ^ {e}frac {{mathrm {partial}tilde {sigma}} ^ {e}} {mathrm {partial}tilde {varepsilon}}tilde {varepsilon}}}mathrm {-}} {tilde {sigma}} ^ {e}frac {partial}tilde {partial}tilde {partial}sigma} _ {mathit {eq}}} ^ {e}} {mathrm {partial}tilde {varepsilon}})

We have:

\[\]

: label: eq-158

{tilde {sigma}} ^ {e}mathrm {=} 2mudeltatilde {varepsilon} +frac {2mu} {{2mu} ^ {mathrm {-}} ^ {mathrm {-}} ^ {mathrm {-}} {mathrm {-}}tofrac {{mathrm {-}} ^ {mathrm {-}}} {mathrm {-}}} {mathrm {-}}}tofrac {{mathrm {-}} ^ {mathrm {-}}}tofrac {{mathrm {-}} ^ {mathrm {-}}}to}tilde {sigma}} ^ {e}} {mathrm {partial}tilde {varepsilon}}}mathrm {=} 2mu I

By Hooke’s law:

\[\]

: label: eq-159

frac {{mathrm {partial}sigma}} _ {mathit {eq}}} ^ {e}} {mathrm {partial}tilde {varepsilon}}}mathrm {=}} 3mathrm {=}} 3mufrac {frac {frac}} {tilde {sigma}} _ {mathit {eq}}}

Finally:

(7.30)\[ \ frac {\ mathrm {\ partial}} {\ mathrm {\ partial}}\ tilde {\ varepsilon}} (\ frac {\ tilde {\ sigma}} {{\ sigma}} _ {\ sigma} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {\ mathit {eq}} _ {}} (2\ mu I\ mathrm {-} 3\ mu\ frac {\ tilde {\ sigma}} {{\ sigma}} _ {\ mathit {eq}}}}\ mathrm {\ otimes}\ frac {\ otimes}\ frac {\ tilde {\ sigma}}} {\ tilde {\ sigma}}})\]

It remains to find the expression for \(\frac{\mathrm{\partial }(\Delta p)}{\mathrm{\partial }\tilde{\varepsilon }}\). To do this, we use the consistency condition \({\overline{f}}^{\text{*}}\mathrm{=}0\):

\[\]

: label: eq-161

{overline {f}} ^ {text {*}}}mathrm {*}}mathrm {=}}frac {{sigma}} ^ {e}mathrm {-} 3muDelta p} 3muDelta p}} {3muDelta p}} {1+3mu {g} _ {Z} _ {Z} ^ {mathit {pt}}}mathrm {-}}mathrm {-} 3muDelta p} {3muDelta p} {1+3muDelta p} {1+3mu {g} _ {Z} _ {Z} ^ {mathit {pt}}}}mathrm {-}}mathrm {-}} 3muDelta p ({p} ^ {mathrm {-}})mathrm {-}}mathrm {-}overline {-} {overline {sigma}} _ {c}mathrm {-}mathrm {-}mathrm {-}overline {-}overline {-}}overline {eta} {eta} (frac {Delta p} {delta t})} ^ {frac {frac {1} {frac {1} {-}overline {-}}mathrm {-}}mathrm {-}}mathrm {-}}mathrm {-}overline {-}overline {-}overline {eta}} {line {n}}}mathrm {=} 0

And we’re drifting away from \(\tilde{\varepsilon }\):

\[\]

: label: eq-162

frac {mathrm {partial} {partial} {overline {f}}} ^ {text {*}}} {mathrm {partial}tilde {varepsilon}}mathrm {=}}frac {overline {f}} {overline {f}} ^ {text {*}}}} {mathrm {partial}}tilde {varepsilon}}left [frac {{sigma} _ {mathit {eq}}} ^ {eq}} ^ {e}}mathrm {-} 3muDelta p} {1+3mu {g} _ {Z}} ^ {mathit {pt}}} {mathit {pt}}}}mathit {pt}}}mathit {pt}}}}mathrm {pt}}}}mathrm {-}}mathrm {-}overline {pt}}}}mathrm {-}}mathrm {-}overline {pt}}}}mathrm {-}overline {pt}}}}mathrm {-}overline {pt}}}}mathrm {-} {Delta p} {Delta t})}}} ^ {frac {1} {overline {n}}}}right]

Thanks to (), we already have \(\frac{{\mathrm{\partial }\sigma }_{\mathit{eq}}^{e}}{\mathrm{\partial }\tilde{\varepsilon }}\). It remains to calculate the other terms. The derivative of the term metallurgical work hardening is equal to:

\[\]

: label: eq-163

frac {mathrm {partial}overline {R}}} {mathrm {partial}tilde {varepsilon}}mathrm {=}overline {{R} _ {R} _ {0}}}frac {mathrm {partial}}frac {mathrm {partial}tilde {0}}}frac {mathrm {partial}tilde {varepsilpsilon}} {mathrm {partial}tilde {varepsilpsilon}} {mathrm {partial}tilde {varepsilpsilon}} no}}

And that of viscous stress:

(7.31)\[ \ frac {\ mathrm {\ partial}} {\ mathrm {\ partial}}\ tilde {\ varepsilon}}\ left [\ overline {\ eta} {(\ frac {\ Delta p} {\ Delta p} {\ Delta p} {\ delta t})}}} {\ frac {\ partial}}\ tilde {\ varepsilon}}\ left [\ overline {\ eta} {(\ frac {\ delta p} {\ delta p} {\ delta p} {\ delta p} {\ delta t})}}} ^ {\ frac {\ delta t})}} ^ {\ frac {\ overline}}}\ right]\ mathrm {=}\ frac {\ overline} {\ eta}} {\ overline {n}\ Delta t} {(\ frac {\ Delta p} {\ Delta t})} ^ {\ frac {1\ mathrm {-}\ overline {n}}} {\ overline {n}}} {\ overline {n}}} {\ overline {n}}} {\ overline {n}}} {\ overline {n}}} {\ overline {n}}} {\ overline {n}}}} (\ frac {\ mathrm {\ partial}}\ tilde {\ varepsilon}})\]

Finally, () has the expression:

\[\]

: label: eq-165

frac {mathrm {partial} {partial} {overline {f}}} ^ {text {*}}} {mathrm {partial}tilde {varepsilon}}mathrm {=}}frac {=}}frac {3mu}} {frac {3mu}} {1+3mu}} {1+3mu {g}} _ {Z} ^ {mathit {pt}}}}mathrm {=}}frac {tilde {sigma}} {{sigma} _ {mathit {pt}}}} {mathit {eq}}}}mathrm {-}frac {3mu {g} _ {Z}} ^ {mathit {pt}}}}frac {mathit {pt}}}}frac {mathrm {partial}tilde {pt}}}}frac {mathrm {partial} (delta p)} {mathrm {partial}tilde {varepsilon}}}frac {mathrm {partial}tilde {varepsilon}}}}mathrm {-}overline {{R} _ {0}}}frac {mathrm {partial} (Delta p)} {mathrm {partial}tilde {varepsilon}}tilde {varepsilon}}}}tilde {varepsilon}}}tilde {varepsilon}}}tilde {varepsilon}}}mathrm {-}frac {frac {Delta p} {Delta t})}}}} ^ {frac {1mathrm {-}overline {n}} {overline {n}}}}} (frac {mathrm {partial} (Delta p)} {mathrm {partial}tilde {varepsilon}})mathrm {=} 0

By rearranging, we find the final expression for \(\frac{\mathrm{\partial }(\Delta p)}{\mathrm{\partial }\tilde{\varepsilon }}\):

\[\]

: label: eq-166

frac {mathrm {partial} (Delta p)} {mathrm {partial}tilde {varepsilon}}mathrm {=}frac {3mu} {3mu} {3mu}} {3mu}} {3mu +left [mu+left] [overline {R} _ {0}}} +frac {overline {eta}}} {overline {n}} {overline {n}}left [overline {R} _ {0}}} +frac {overline {eta}}} {overline {n}}Delta t} {(frac {Delta p} {Delta p} {Delta p} {Delta t})}} ^ {frac {1mathrm {-}}overline {n}}}right]right]right]leftleft [1+3mu {g}}left [1+3mu {g}} _ {g} _ {Z} _ {Z} ^ {mathit {pt}}right]}frac {tilde {sigma}}right]}right]left [1+3mu {g}} _ {g} _ {G} _ {Z} ^ {mathit {pt}}right]}frac {tilde {sigma}}right]}right]right]left sigma} _ {mathit {eq}}}

So for the term visco-plastic:

\[\]

: label: eq-167

frac {{mathrm {partial}tilde {varepsilon}}} ^ {mathit {VP}}} {mathrm {partial}tilde {varepsilon}}mathrm {=}mathrm {=}}mathrm {=}}frac {3} {2}left [frac {3mu} {3mu} {3mu} {3mu +left [overline {{R} _ {0}} +frac {frac {frac {1overline {eta}}} {overline {n}Delta t} {delta t})} ^ {frac {1frac {1mathrm {1mathrm {-}}mathrm {-}}overline {n}}} {overline {n}}} {frac {delta t})}} ^ {frac {1frac {1mathrm {-} -}mathrm {-}mathrm {-}overline {n}}} {overline {n}}} {overline {n}}}right]left [1+3mu {g} _ {Z} ^ {mathit {pt}}right]}right]}frac {tilde {sigma}} {{sigma} _ {mathit {eq}}}mathrm {otimes}frac {tilde {sigma}}} {tilde {sigma}}}} {mathit {eq}}}}mathrm {otimes}frac {1} {sigma}}} +Delta pfrac {1} {sigma}} +Delta pfrac {1} {sigma} _ {mathit {eq}} ^ {e}} (2mu Imathrm {-} 3mufrac {tilde {sigma}} {{sigma}} {{sigma}} _ {sigma}} _ {mathit {eq}}}mathrm {otimes}frac {tilde {sigma}} {{sigma} _ {mathit {eq}}}})right]

Starting from the formal expression of the deviatoric tangent matrix:

\[\]

: label: eq-168

frac {mathrm {partial}tilde {sigma}} {mathrm {partial}tilde {varepsilon}}mathrm {=}frac {mathrm {partial}}} {mathrm {partial}} {mathrm {partial}}tilde {varepsilon}}left [2mu (tilde {partial}}left [2mu (tilde {partial}} arepsilon}mathrm {-} {tilde {varepsilon}}} ^ {mathit {VP}}mathrm {-} {tilde {varepsilon}} {tilde {varepsilon}}}} ^ {mathit {pt}})right]

It is injected with the expression of the term corresponding to transformation plasticity and elasticity:

\[\]

: label: eq-169

frac {mathrm {partial}tilde {sigma}} {mathrm {partial}tilde {varepsilon}}mathrm {=} 2mu Imathrm {-} 2mumathrm {-} 2mufrac {frac {mathrm {partial}tilde {varepsilon}}} ^ {mathit {VP}} ^ {mathit {VP}}} {mathrm {partial}tilde {varepsilon}}mathrm {-} 2mufrac {3} {2} {g} _ {Z} ^ {mathit {pt}}tilde {mathit {pt}}}tilde {mathrm {partial}}tilde {partial}tilde {partial}tilde {varepsilon}}tilde {varepsilon}}tilde {varepsilon}}tilde {varepsilon}}tilde {varepsilon}}tilde {varepsilon}}tilde {varepsilon}}tilde {varepsilon Silon}}

And so:

\[\]

: label: eq-170

frac {mathrm {partial}tilde {sigma}} {mathrm {partial}tilde {varepsilon}}mathrm {=}frac {1} {1} {1+3mu {1} {1+3mu {g}} {1+3mu {g}}} {1+3mu {g}} _ {Z} _ {Z}} ^ {mathit {pt}}}}left [2mu Imathrm {-} 1} {1} {1+3mu {g}} {1+3mu {g}} _ {Z}} _ {Z}} ^ {mathit {pt}}}}left [2mu Imathrm {-} 1} {1} {ac {{mathrm {partial}tilde {varepsilon}}} ^ {mathit {VP}}} {mathrm {partial}tilde {varepsilon}}right]

The deviatoric tangent matrix is broken down into two terms:

\[\]

: label: eq-171

frac {mathrm {partial}tilde {sigma}} {mathrm {partial}tilde {varepsilon}}mathrm {=}frac {1} {1} {1+3mu {g} {1+3mu {g}}} {mu {g}}} {mathrm {partial}}}mathrm {=}frac {1} {1} {1+3mu {g}} _ {Z} _ {Z} _ {Z} ^ {mathit {pt}}}}left [alpha I+betafrac {1}} {1} {1+3mu {g}} _ {G} _ {Z} _ {Z}} ^ {mathit {pt}}} {{sigma} _ {mathit {eq}}}}}mathrm {otimes}frac {tilde {sigma}} {{sigma}} _ {mathit {eq}}}right]

With the term \(\alpha\):

\[\]

: label: eq-172

alphamathrm {=} 2mu (1mathrm {-}frac {3muDelta p} {{sigma}} _ {mathit {eq}}} ^ {e}})

And the term \(\beta\):

\[\]

: label: eq-173

betamathrm {=}mathrm {-} {9mu}} {9mu} ^ {2}left [frac {1} {3mu +left [overline {{R} _ {0}}} +frac {overline {- 0}}} +frac {overline {eta}}} {overline {eta}}} {overline {eta}}} {overline {eta}}} {overline {eta}}} {overline {eta}}} {overline {n}delta t} {delta t}} {(frac {Delta p} _ {0}}}} +frac {overline {0}}}} ^ {frac {1mathrm {-}overline {n}} {overline {n}}}}right]left [1+3mu {g} _ {Z} ^ {mathit {pt}}}right}}right]}right]} +frac {delta p}} {delta p} {sigma} _ {mathit {eq}} {Z} ^ {e}}}right]

7.5. Synthesis of matrices

The prediction and correction matrices can be written in the same form, by varying the coefficients as appropriate. The spherical part is the same in both matrices. The deviatoric part is written in general form:

\[\]

: label: eq-174

tilde {K}mathrm {=}left [frac {2mu} {a} (1mathrm {-} {c} _ {3}frac {3muDelta p} {{delta p}} {{sigma}} _ {sigma}} _ {p} _ {p} _ {p}} _ {p}}) Imathrm {-}frac {{c} _ {p}} {a}tilde {sigma}mathrm {otimes}tilde {sigma}right]

With the following table for the coefficients:

	RIGI_MECA_TANG
\(a\)	\(1\)
\({c}_{3}\)	\(0\)
\({c}_{p}\)	\({c}_{1}\frac{{(3\mu )}^{2}}{{({\sigma }_{\mathit{eq}}^{e})}^{2}}\left[\frac{1}{3\mu +\overline{{R}_{0}}}\right]\)
\({c}_{1}\)	is 1 if laminating, 0 otherwise
\({c}_{2}\)	is 1 if we charge, 0 otherwise

	FULL_MECA
\(a\)	\(1+3\mu {g}_{Z}^{\mathit{pt}}\)
\({c}_{3}\)	\(1\)
\({c}_{p}\)	\({c}_{2}\frac{{(3\mu )}^{2}}{{({\sigma }_{\mathit{eq}}^{e})}^{2}}\left[\frac{1}{3\mu +\left[\overline{{R}_{0}}+\frac{\overline{\eta }}{\overline{n}\Delta t}{(\frac{\Delta p}{\Delta t})}^{\frac{1\mathrm{-}\overline{n}}{\overline{n}}}\right]\left[1+3\mu {g}_{Z}^{\mathit{pt}}\right]}\mathrm{-}\frac{\Delta p}{{\sigma }_{\mathit{eq}}^{e}}\right]\)
\({c}_{1}\)	is 1 if laminating, 0 otherwise
\({c}_{2}\)	is 1 if we charge, 0 otherwise

[1]

A command variable is a scalar quantity, a function of time and space, given a priory by the user via the AFFE_VARC keyword in the AFFE_MATERIAU operator. It is a parameter of the problem and not an unknown.

[2]

Which inverts the sign in front of this contribution!