4. Integration of the CJS law#

Below we detail the integration of the CJS law according to the mechanism (s) activated:

  • non-linear elastic,

  • nonlinear elastic and isotropic plastic

  • nonlinear elastic and deviatory plastic

  • nonlinear elastic, isotropic plastic and deviatory plastic.

In each case, the aim is to calculate, from the fields known in the minus state \({\mathrm{\varepsilon }}^{-}\), \({\sigma }^{-}\) and the deformation increment \(\Delta \varepsilon\), the new stress state \({\sigma }^{+}\).

In the sequence of calculations, we start by making the assumption that only the nonlinear elastic mechanism is involved. An elastic prediction is therefore carried out. This prediction is then used to calculate the load functions \({f}^{i}\) and \({f}^{d}\), we want to know if we then go beyond the thresholds:

  • if \({f}^{i}\le 0\) and \({f}^{d}\le 0\), elastic prediction is retained as the new stress state,

  • if \({f}^{i}>0\) and \({f}^{d}\le 0\), we integrate the nonlinear elastic and isotropic plastic mechanisms,

  • if \({f}^{i}\le 0\) and \({f}^{d}>0\), we integrate the nonlinear elastic and deviatory plastic mechanisms,

  • if \({f}^{i}>0\) and \({f}^{d}>0\), we integrate the mechanisms of nonlinear elastic, isotropic plastic and deviatory plastic.

At the end of the elasto-plastic calculation, when a single plastic threshold has been initially exceeded, each of the load functions is recalculated. In fact, it is possible that by seeking to return to one of the thresholds, one then exceeds the other threshold not initially activated by the elastic prediction. In this case, the solution is then solved by integrating all the mechanisms.

4.1. Choice of internal variables#

The variables \(q\), \(r\), and \(\alpha\) are equivalent to the associated thermodynamic forces \({Q}_{\text{iso}}\), \(R\), and \(X\). For this reason and since their geometric meaning is more obvious, we will use as internal variables for the integration of law CJS, the quantities \({Q}_{\text{iso}}\), \(R\) and \(X\).

In addition, we add to the number of internal variables:

  • the product sign \({s}_{\text{ij}}{\mathrm{\varepsilon }}_{\text{ij}}^{\text{dp}}\)

  • the elastic or elasto-plastic state of the material, noting:

  • 0: elastic state

  • 1: elasto-plastic state, isotropic plastic mechanism

  • 2: elasto-plastic state, deviatory plastic mechanism

  • 3: elasto-plastic state, isotropic and deviatory plastic mechanisms

Finally, the internal variables are stored in a VI vector in the following order:

Internal variable index

CJS1

CJS2

CJS3

3D

2D

CJS1

CJS2

CJS3

1

1

\({Q}_{\text{iso}}=\infty\)

\({Q}_{\text{iso}}\)

\({Q}_{\text{iso}}\)

2

2

\(R={R}_{m}\)

\(R\)

\(R={R}_{m}\)

3

3

0

0

\({X}_{\text{11}}\)

4

4

0

0

\({X}_{\text{22}}\)

5

5

0

0

\({X}_{\text{33}}\)

6

6

0

0

\(\sqrt{2}{X}_{\text{12}}\)

7

—

0

0

\(\sqrt{2}{X}_{\text{13}}\)

8

—

0

0

\(\sqrt{2}{X}_{\text{23}}\)

9

7

\(\frac{{q}_{\text{II}}h({\theta }_{q})}{\mid {R}_{m}({I}_{1}+{Q}_{\text{init}})\mid }\)

\(\frac{{q}_{\text{II}}h({\theta }_{q})}{\mid R({I}_{1}+{Q}_{\text{init}})\mid }\)

\(\frac{{q}_{\text{II}}h({\theta }_{q})}{\mid {R}_{m}({I}_{1}+{Q}_{\text{init}})\mid }\)

10

8

\(\frac{R}{{R}_{m}}\)

\(\frac{{X}_{\text{II}}}{{X}_{\text{II}}^{\text{lim}}}\)

11

9

\(\mid \frac{\mathrm{3Q}}{{I}_{1}+{Q}_{\text{init}}}\mid\)

\(\mid \frac{\mathrm{3Q}}{{I}_{1}+{Q}_{\text{init}}}\mid\)

12

10

Number of internal iterations

Number of internal iterations

Number of internal iterations

13

11

Local test reached

Local test reached

Local test reached

14

12

Number of redistricting

Number of redistricting

Number of redistricting

15

13

\(\text{signe}({s}_{\text{ij}}{\varepsilon }_{\text{ij}}^{\text{dp}})\)

\(\text{signe}({s}_{\text{ij}}{\varepsilon }_{\text{ij}}^{\text{dp}})\)

\(\text{signe}({s}_{\text{ij}}{\varepsilon }_{\text{ij}}^{\text{dp}})\)

16

14

0,1,2,3 state of the material

0,1,2,3 state of the material

0,1,2,3 state of the material

4.2. Integration of the nonlinear elastic mechanism#

In the elastic case, the new stress state \({\sigma }^{+}\) simply checks:

\({\sigma }_{\text{ij}}^{+}={\sigma }_{\text{ij}}^{-}+{D}_{\text{ijkl}}({\sigma }^{\text{+}})\Delta {\varepsilon }_{\text{kl}}\)

The dependence of the nonlinear elasticity tensor on the stress state actually boils down to:

\({D}_{\text{ijkl}}({\sigma }^{+})={D}_{\text{ijkl}}^{\text{lineaire}}{(\frac{{I}_{1}^{+}+{Q}_{\text{init}}}{3{P}_{a}})}^{n}\)

where \({D}_{\text{ijkl}}^{\text{lineaire}}\) is the classical isotropic linear elasticity tensor, obtained from \({K}_{o}^{e}\) and \({G}_{o}\) or by equivalence from E and Nu.

From this relationship, it is deduced in particular that the first invariant of the constraints satisfies:

\({I}_{1}^{+}-{I}_{1}^{-}-3{K}_{o}^{e}{(\frac{{I}_{1}^{+}+{Q}_{\text{init}}}{3{P}_{a}})}^{n}\text{tr}(\Delta \varepsilon )=0\)

This nonlinear equation is solved by a secant method for CJS2 and CJS3, by differentiating the cases according to the sign of \(\text{tr}(\mathrm{\Delta }\mathrm{\varepsilon })\). With regard to model CJS1, for which the parameter \(n\) is zero, the explicit resolution is immediate, since we then have

\({I}_{1}^{+}={I}_{1}^{-}+3{K}_{o}^{e}\text{tr}(\Delta \varepsilon )\)

In the general case, knowledge of \({I}_{1}^{+}\) and therefore of the term \({(\frac{{I}_{1}^{+}+{Q}_{\text{init}}}{3{P}_{a}})}^{n}\) makes it possible to define the nonlinear elasticity operator \({D}_{\text{ijkl}}({\sigma }^{+})\). The new state of constraint is then obtained directly.

4.3. Integration of nonlinear elastic and isotropic plastic mechanisms#

In this case, the new constraint state \({\sigma }^{+}\) checks:

\({\sigma }_{\text{ij}}^{+}={\sigma }_{\text{ij}}^{-}+{D}_{\text{ijkl}}({\sigma }^{+})(\Delta {\varepsilon }_{\text{kl}}-\Delta {\varepsilon }_{\text{kl}}^{\text{ip}})\)

Given the simple shape, plastic deformations of the isotropic plastic mechanism:

\(\Delta {\varepsilon }_{\text{ij}}^{\text{ip}}=-\frac{1}{3}\Delta {\lambda }^{i}{\delta }_{\text{ij}}\)

the nonlinear system to be solved is composed of:

  • \({\text{LE}}_{\text{ij}}\): the elastic state law: \({\sigma }_{\text{ij}}^{+}-{\sigma }_{\text{ij}}^{-}-{D}_{\text{ijkl}}({\sigma }^{+})(\Delta {\varepsilon }_{\text{kl}}+\frac{1}{3}\Delta {\lambda }^{i}{\delta }_{\text{kl}})=0\)

  • \(\text{LQ}\): the law of hardening of the internal variable \({Q}_{\text{iso}}\): \({Q}_{\text{iso}}^{+}-{Q}_{\text{iso}}^{-}-\Delta {\lambda }^{i}{G}^{{Q}_{\text{iso}}}({Q}_{\text{iso}}^{+})=0\)

  • \(\text{FI}\): the isotropic charge area equation: \(-\frac{{I}_{1}^{+}+{Q}_{\text{init}}}{3}+{Q}_{\text{iso}}^{+}=0\)

Schematically, we therefore seek to solve the system \(R(Y)=0\), where the unknown \(Y\) is given by \(Y=({\sigma }_{\text{ij}}^{+},{Q}_{\text{iso}}^{+},\Delta {\lambda }^{i})\) and where \(R=({\text{LE}}_{\text{ij}},\text{LQ},\text{FI})\). The resolution of \(R(Y)=0\) is done by Newton’s method:

  • initialization and calculation of a test solution

  • Newton iterations: solving \(\frac{\text{DR}}{\text{DY}}({Y}^{p}){\text{DY}}^{\text{p+1}}=-R({Y}^{p})\)

  • convergence test: if convergence \(Y={Y}^{p}\); otherwise \({Y}^{\text{p+1}}={Y}^{p}+{\text{DY}}^{\text{p+1}}\) and \(p=p+1\)

We detail these three steps below.

4.3.1. Initialization and test solution#

We simply take the following values for \({Y}^{0}=({\sigma }_{\text{ij}}^{0},{Q}_{\text{iso}}^{0},\Delta {\lambda }^{{i}^{0}})\):

\({\sigma }_{\text{ij}}^{0}={\sigma }_{\text{ij}}^{\text{elas}}\): constraints given by elastic prediction,

\({Q}_{\text{iso}}^{0}={Q}_{\text{iso}}^{-}\): variable internal to t

\(\Delta {\lambda }^{{i}^{0}}=0\): zero plastic multiplier

Unlike other elasto-plastic mechanisms, a test solution is not calculated here.

4.3.2. Newton iterations#

Solving \(\frac{\text{DR}}{\text{DY}}({Y}^{p}){\text{DY}}^{\text{p+1}}=-R({Y}^{p})\) naturally requires calculating the derivatives of \({\text{LE}}_{\text{ij}}\), \(\text{LQ}\), and \(\text{FI}\) with respect to each component of \(Y\). We have:

\(\frac{\text{DR}}{\text{DY}}=\left[\begin{array}{ccc}\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\sigma }_{\text{kl}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {Q}_{\text{iso}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial \Delta {\lambda }^{i}}\\ \frac{\partial \text{LQ}}{\partial {\sigma }_{\text{kl}}}& \frac{\partial \text{LQ}}{\partial {Q}_{\text{iso}}}& \frac{\partial \text{LQ}}{\partial \Delta {\lambda }^{i}}\\ \frac{\partial \text{FI}}{\partial {\sigma }_{\text{kl}}}& \frac{\partial \text{FI}}{\partial {Q}_{\text{iso}}}& \frac{\partial \text{FI}}{\partial \Delta {\lambda }^{i}}\end{array}\right]\)

with:

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\sigma }_{\text{kl}}\delta }={\delta }_{\text{ik}}{\delta }_{\text{jl}}-\frac{\partial {D}_{\text{ijmn}}}{\partial {\sigma }_{\text{kl}}}(\Delta {\epsilon }_{\text{mn}}+\frac{1}{3}\Delta {\lambda }^{i}{\delta }_{\text{mn}})={\delta }_{\text{ik}}{\delta }_{\text{jl}}-{D}_{\text{ijmn}}^{\text{lineaire}}(\Delta {\epsilon }_{\text{mn}}+\frac{1}{3}\Delta {\lambda }^{i}{\delta }_{\text{mn}})\frac{n}{3{P}_{a}}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{n-\delta }{\delta }_{\text{kl}}\)

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {Q}_{\text{iso}}}=0\)

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial \Delta {\lambda }^{i}}=-\frac{1}{3}{D}_{\text{ijmn}}{\delta }_{\text{mn}}\)

\(\frac{\partial \text{LQ}}{\partial {\sigma }_{\text{kl}}}=0\)

\(\frac{\partial \text{LQ}}{\partial {Q}_{\text{iso}}}=1-\Delta {\lambda }^{i}\frac{\partial {G}^{{Q}_{\text{iso}}}}{\partial {Q}_{\text{iso}}}=1+\Delta {\lambda }^{i}\frac{n{K}_{o}^{p}}{{P}_{a}}{(\frac{{Q}_{\text{iso}}}{{P}_{a}})}^{n-1}\)

\(\frac{\partial \text{LQ}}{\partial \Delta {\lambda }^{i}}=-{G}^{{Q}_{\text{iso}}}\)

\(\frac{\partial \text{FI}}{\partial {\sigma }_{\text{kl}}}=-\frac{1}{3}{\delta }_{\text{kl}}\)

\(\frac{\partial \text{FI}}{\partial {Q}_{\text{iso}}}=1\)

\(\frac{\partial \text{FI}}{\partial \Delta {\lambda }^{i}}=0\)

4.3.3. Convergence test#

Newton’s iterations are continued as long as the relative error \(\frac{\parallel {\text{DY}}^{\text{p+1}}\parallel }{\parallel {Y}^{\text{p+1}}-{Y}^{0}\parallel }\) remains greater than the tolerance accepted by the user and defined by the keyword RESI_INTE_RELA. The standard used here is the vector norm: \(\parallel x\parallel =\sqrt{\sum _{i}{x}_{i}^{2}}\).

4.4. Integration of nonlinear elastic and deviatory plastic mechanisms#

In this case, the new constraint state \({\sigma }^{+}\) checks:

\({\sigma }_{\text{ij}}^{+}={\sigma }_{\text{ij}}^{-}+{D}_{\text{ijkl}}({\sigma }^{+})(\Delta {\varepsilon }_{\text{kl}}-\Delta {\varepsilon }_{\text{kl}}^{\text{dp}})\)

The plastic deformations of the deviatory plastic mechanism are given by the potential \({G}^{d}\):

\(\Delta {\varepsilon }_{\text{ij}}^{\text{dp}}=\Delta {\lambda }^{d}{G}_{\text{ij}}^{d}\)

From this we deduce that the nonlinear system to be solved is composed of:

  • \({\text{LE}}_{\text{ij}}\): the elastic state law: \({\sigma }_{\text{ij}}^{+}-{\sigma }_{\text{ij}}^{-}-{D}_{\text{ijkl}}({\sigma }^{+})(\Delta {\varepsilon }_{\text{kl}}-\Delta {\lambda }^{d}{G}_{\text{kl}}^{d}({\sigma }^{+},{R}^{+},{X}^{+}))=0\)

  • \(\text{LR}\): the law of hardening of the variable \(R\): \({R}^{+}-{R}^{-}-{\Delta }^{d}{G}^{R}({\sigma }^{+},{R}^{+})=0\)

  • \({\text{LX}}_{\text{ij}}\): the law of hardening of the variable \({X}_{\text{ij}}\): \({X}_{\text{ij}}^{+}-{X}_{\text{ij}}^{-}-\Delta {\lambda }^{d}{G}^{X}({\mathrm{\sigma }}^{+},{X}^{+})=0\)

  • \(\text{FD}\): the equation of the deviatory load area: \({q}_{\text{II}}^{+}h({\mathrm{\theta }}_{q}^{+})+{R}^{+}({I}_{1}^{+}+{Q}_{\text{init}})=0\)

As in the preceding paragraph, system \(R(Y)=0\) is solved by Newton’s method, where the unknown \(Y\) is given by \(Y=({\sigma }_{\text{ij}}^{+},{R}^{+},{X}_{\text{ij}}^{+},\Delta {\lambda }^{d})\) and where \(R=({\text{LE}}_{\text{ij}},\text{LR},{\text{LX}}_{\text{ij}},\text{FD})\).

4.4.1. Initialization and test solution#

Starting with the state at moment t \(({\sigma }_{\text{ij}}^{-},{R}^{-},{X}_{\text{ij}}^{-})\), we are looking for a test solution that brings us closer to the final solution. To do this we solve the following equation:

\({f}^{d}({\sigma }_{\text{ij}}^{-}+{D}_{\text{ijkl}}^{-}(\Delta {\varepsilon }_{\text{kl}}-\Delta {\lambda }^{d}{G}_{\text{kl}}^{d-}),{R}^{-}+\Delta {\lambda }^{d}{G}^{R-},{X}_{\text{ij}}^{-}+\Delta {\lambda }^{d}{G}_{\text{ij}}^{X-})=0\)

with \({D}_{\text{ijkl}}^{-}={D}_{\text{ijkl}}({\sigma }^{-})\), \({G}_{\text{kl}}^{d-}={G}_{\text{kl}}^{d}({\sigma }^{-},{R}^{-},{X}^{-})\), \({G}^{R-}={G}^{R}({\sigma }^{-},{R}^{-})\), \({G}_{\text{ij}}^{X-}={G}_{\text{ij}}^{X}({\sigma }^{-},{X}^{-})\) and where the unknown is the plastic multiplier \(\Delta {\lambda }^{d}\), by a single Newton’s iteration, that is to say finally of we have:

\({\frac{\partial {f}^{d}}{\partial \Delta {\lambda }^{d}}}_{\mid \Delta {\lambda }^{d}=0}\Delta {\lambda }^{d}=-{{f}^{d}}_{\mid \Delta {\lambda }^{d}=0}\) or \(\Delta {\lambda }^{d}=-\frac{{f}_{\mid \Delta {\lambda }^{d}=0}^{d}}{{\frac{\partial {f}^{d}}{\partial \Delta {\lambda }^{d}}}_{\mid \Delta {\lambda }^{d}=0}}\) again

with:

\(\frac{\partial {f}^{d}}{\partial \Delta {\lambda }^{d}}=h({\theta }_{q})\frac{\partial {q}_{\text{II}}}{\partial \Delta {\lambda }^{d}}+{q}_{\text{II}}\frac{\partial h({\theta }_{q})}{\partial \Delta {\lambda }^{d}}+({I}_{1}+{Q}_{\text{init}})\frac{\partial R}{\partial \Delta {\lambda }^{d}}+R\frac{\partial {I}_{1}}{\partial \Delta {\lambda }^{d}}\)

In addition,

We have: \({I}_{1}={I}_{1}^{-}+3{K}^{-}(\text{tr}(\Delta \varepsilon )-\Delta {\lambda }^{d}\text{tr}({G}^{d-}))\) then: \(\frac{\partial {I}_{1}}{\partial \Delta {\lambda }^{d}}=-3{K}^{-}\text{tr}({G}^{d-})\)

We have: \(R={R}^{-}+\Delta {\lambda }^{d}{G}^{R-}\) then: \(\frac{\partial R}{\partial \Delta {\lambda }^{d}}={G}^{R-}\)

we have:

\({q}_{\text{ij}}={\mathrm{\sigma }}_{\text{ij}}^{-}+{D}_{\text{ijkl}}^{-}({\mathrm{\Delta \varepsilon }}_{\text{kl}}-{\mathrm{\Delta \lambda }}^{d}{G}_{\text{kl}}^{d-})-\left[{I}_{1}^{-}+3{K}^{-}(\text{tr}(\mathrm{\Delta \varepsilon })-{\mathrm{\Delta \lambda }}^{d}\text{tr}({G}^{d-}))\right]\left[\frac{1}{3}{\mathrm{\delta }}_{\text{ij}}+{X}_{\text{ij}}^{-}+{\mathrm{\Delta \lambda }}^{d}{G}_{\text{ij}}^{X-}\right]\)

So: \({\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}}_{\mid {\mathrm{\Delta \lambda }}^{d}=0}=-{D}_{\text{ijkl}}^{-}{G}_{\text{kl}}^{d-}+3{K}^{-}\text{tr}({G}^{d-})(\frac{1}{3}{\mathrm{\delta }}_{\text{ij}}+{X}_{\text{ij}}^{-})-{G}_{\text{ij}}^{X-}({I}_{1}^{-}+3{K}^{-}\text{tr}(\mathrm{\Delta }\mathrm{\varepsilon }))\)

We have: \(\frac{\partial {q}_{\text{II}}}{\partial {\mathrm{\Delta \lambda }}^{d}}=\frac{\partial {q}_{\text{II}}}{\partial {q}_{\text{ij}}}\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}=\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}\) and \(\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {\mathrm{\Delta \lambda }}^{d}}=\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {q}_{\text{ij}}}\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}\)

In short, we take \({Y}^{0}=({\mathrm{\sigma }}_{\text{ij}}^{0},{R}^{0},{X}_{\text{ij}}^{0},{\mathrm{\Delta \lambda }}^{\mathrm{d0}})\) for the test solution, with the following values:

\({\mathrm{\Delta \lambda }}^{\mathrm{d0}}\): the value found according to the previous formulation.

\({\mathrm{\sigma }}_{\text{ij}}^{0}={\mathrm{\sigma }}_{\text{ij}}^{-}+{D}_{\text{ijkl}}^{-}({\mathrm{\Delta \varepsilon }}_{\text{kl}}-{\mathrm{\Delta \lambda }}^{\mathrm{d0}}{G}_{\text{kl}}^{d-})\)

\({R}^{0}={R}^{-}+{\mathrm{\Delta \lambda }}^{\mathrm{d0}}{G}^{R-}\)

\({X}_{\text{ij}}^{0}={X}_{\text{ij}}^{-}+{\mathrm{\Delta \lambda }}^{\mathrm{d0}}{G}_{\text{ij}}^{X-}\)

4.4.2. Newton iterations#

\(\frac{\text{DR}}{\text{DY}}\) is here given by:

\(\frac{\text{DR}}{\text{DY}}=\left[\begin{array}{cccc}\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial R}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {X}_{\text{ij}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial \text{LR}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial \text{LR}}{\partial R}& \frac{\partial \text{LR}}{\partial {X}_{\text{ij}}}& \frac{\partial \text{LR}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial R}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {X}_{\text{ij}}}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial \text{FD}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial \text{FD}}{\partial R}& \frac{\partial \text{FD}}{\partial {X}_{\text{ij}}}& \frac{\partial \text{FD}}{\partial {\mathrm{\Delta \lambda }}^{d}}\end{array}\right]\)

with:

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}={\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}}-{D}_{\text{ijmn}}^{\text{lineaire}}({\mathrm{\Delta \varepsilon }}_{\text{mn}}-{\mathrm{\Delta \lambda }}^{d}{G}_{\text{mn}}^{d})\frac{n}{3{P}_{a}}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{n-1}{\mathrm{\delta }}_{\text{kl}}+{D}_{\text{ijmn}}{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial R}={D}_{\text{ijmn}}{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}_{\text{mn}}^{d}}{\partial R}\)

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {X}_{\text{kl}}}={D}_{\text{ijmn}}{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}_{\text{mn}}^{d}}{\partial {X}_{\text{kl}}}\)

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}={D}_{\text{ijmn}}{G}_{\text{mn}}^{d}\)

\(\frac{\partial \text{LR}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}^{R}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-{\mathrm{\Delta \lambda }}^{d}\frac{A}{2}{(1-\frac{R}{{R}_{m}})}^{2}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}{\mathrm{\delta }}_{\text{kl}}\)

\(\frac{\partial \text{LR}}{\partial R}=1-{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}^{R}}{\partial R}=1-{\mathrm{\Delta \lambda }}^{d}\frac{2A}{{R}_{m}}(1-\frac{R}{{R}_{m}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\)

\(\frac{\partial \text{LR}}{\partial {X}_{\text{kl}}}=0\)

\(\frac{\partial \text{LR}}{\partial {\mathrm{\Delta \lambda }}^{d}}=-{G}^{R}\)

\(\frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}_{\text{ij}}^{X}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(\frac{\partial {\text{LX}}_{\text{ij}}}{\partial R}=0\)

\(\frac{\partial {\text{LX}}_{\text{ij}}}{\partial {X}_{\text{kl}}}={\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}}-{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}_{\text{ij}}^{X}}{\partial {X}_{\text{kl}}}\)

\(\frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}=-{G}_{\text{ij}}^{X}\)

\(\frac{\partial \text{FD}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}={Q}_{\text{kl}}-({Q}_{\text{mn}}{X}_{\text{mn}}-R){\mathrm{\delta }}_{\text{kl}}\)

\(\frac{\partial \text{FD}}{\partial R}={I}_{1}\)

\(\frac{\partial \text{FD}}{\partial {X}_{\text{kl}}}=\frac{\partial {f}^{d}}{\partial {X}_{\text{kl}}}\)

\(\frac{\partial \text{FD}}{\partial {\mathrm{\Delta \lambda }}^{d}}=0\)

In addition, the calculation of the terms \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\), \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial R}\), \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {X}_{\text{kl}}}\),, \(\frac{\partial {G}_{\text{ij}}^{X}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\),, \(\frac{\partial {G}_{\text{ij}}^{X}}{\partial {X}_{\text{kl}}}\) and \(\frac{\partial {f}^{d}}{\partial {X}_{\text{kl}}}\) is detailed below, as well as the calculation of useful intermediate terms:

  • calculation of \(\frac{\partial {f}^{d}}{\partial {X}_{\text{kl}}}\):

\(\frac{\partial {f}^{d}}{\partial {X}_{\text{kl}}}={q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {X}_{\text{kl}}}+h({\mathrm{\theta }}_{q})\frac{\partial {q}_{\text{II}}}{\partial {X}_{\text{kl}}}\)

\(={q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {q}_{\text{mn}}}\frac{\partial {q}_{\text{mn}}}{\partial {X}_{\text{kl}}}+h({\mathrm{\theta }}_{q})\frac{\partial {q}_{\text{II}}}{\partial {q}_{\text{mn}}}\frac{\partial {q}_{\text{mn}}}{\partial {X}_{\text{kl}}}\)

\(={q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {q}_{\text{mn}}}\frac{\partial {q}_{\text{mn}}}{\partial {X}_{\text{kl}}}+h({\mathrm{\theta }}_{q})\frac{{q}_{\text{mn}}}{{q}_{\text{II}}}\frac{\partial {q}_{\text{mn}}}{\partial {X}_{\text{kl}}}\)

\(=({q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {q}_{\text{mn}}}+h({\mathrm{\theta }}_{q})\frac{{q}_{\text{mn}}}{{q}_{\text{II}}})\frac{\partial {q}_{\text{mn}}}{\partial {X}_{\text{kl}}}\)

\(=-{I}_{1}({q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {q}_{\text{mn}}}+h({\mathrm{\theta }}_{q})\frac{{q}_{\text{mn}}}{{q}_{\text{II}}}){\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}\)

\(=-{I}_{1}(\frac{\partial {f}^{d}}{\partial {q}_{\text{kl}}})\)

It will be noted below that:

\(\text{dev}(\frac{\partial {f}^{d}}{\partial {X}_{\text{kl}}})=-{I}_{1}{Q}_{\text{kl}}\)

  • calculation of \(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{ij}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{ij}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{ij}}})}{\partial {q}_{\text{mn}}}\frac{\partial {q}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=(\frac{\partial ({Q}_{\text{ij}}-({Q}_{\text{rs}}{X}_{\text{rs}}-R){\mathrm{\delta }}_{\text{ij}})}{\partial {q}_{\text{mn}}})\frac{\partial {q}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=(\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}-(\frac{\partial {Q}_{\text{rs}}}{\partial {q}_{\text{mn}}}{X}_{\text{rs}}){\mathrm{\delta }}_{\text{ij}})\frac{\partial {q}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=(\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}-(\frac{\partial {Q}_{\text{rs}}}{\partial {q}_{\text{mn}}}{X}_{\text{rs}}){\mathrm{\delta }}_{\text{ij}})({\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}-{\mathrm{\delta }}_{\text{kl}}(\frac{{\mathrm{\delta }}_{\text{mn}}}{3}+{X}_{\text{mn}}))\)

  • calculation of \(\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}\):

Beforehand, we define the tensor \(t\) and its deviatory part \({t}^{d}\) by setting:

\({t}_{\text{ij}}=\frac{\partial \text{det}(q)}{\partial {q}_{\text{ij}}}\) and \({t}_{\text{ij}}^{d}=\text{dev}(\frac{\partial \text{det}(q)}{\partial {q}_{\text{ij}}})\)

We thus have:

\(t=\left[\begin{array}{c}{t}_{\text{11}}\\ {t}_{\text{22}}\\ {t}_{\text{33}}\\ {t}_{\text{12}}\\ {t}_{\text{13}}\\ {t}_{\text{23}}\end{array}\right]=\left[\begin{array}{c}{q}_{\text{22}}{q}_{\text{33}}-{q}_{\text{23}}{q}_{\text{23}}\\ {q}_{\text{11}}{q}_{\text{33}}-{q}_{\text{13}}{q}_{\text{13}}\\ {q}_{\text{11}}{q}_{\text{22}}-{q}_{\text{12}}{q}_{\text{12}}\\ {q}_{\text{13}}{q}_{\text{23}}-{q}_{\text{12}}{q}_{\text{33}}\\ {q}_{\text{12}}{q}_{\text{23}}-{q}_{\text{13}}{q}_{\text{22}}\\ {q}_{\text{12}}{q}_{\text{13}}-{q}_{\text{23}}{q}_{\text{11}}\end{array}\right]\)

\(\begin{array}{}\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}=\frac{-5}{h{({\theta }_{q})}^{6}}\left[(1+\frac{\gamma }{2}\text{cos}(3{\theta }_{q}))\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}+\frac{\gamma \sqrt{\text{54}}}{6{q}_{\text{II}}^{2}}\text{dev}({t}_{\text{ij}})\right]\frac{\partial (h({\theta }_{q}))}{\partial {q}_{\text{mn}}}\\ +\frac{1}{h{({\theta }_{q})}^{5}}(1+\frac{\gamma }{2}\text{cos}(3{\theta }_{q}))\frac{\partial (\frac{{q}_{\text{ij}}}{{q}_{\text{II}}})}{\partial {q}_{\text{mn}}}+\frac{1}{h{({\theta }_{q})}^{5}}\frac{\gamma }{2}\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}\frac{\partial \text{cos}(3{\theta }_{q})}{\partial {q}_{\text{mn}}}+\frac{1}{h{({\theta }_{q})}^{5}}\frac{\sqrt{\text{54}}\gamma }{6}\frac{\partial (\frac{{t}_{\text{ij}}^{d}}{{q}_{\text{II}}^{2}})}{\partial {q}_{\text{mn}}}\\ =\frac{-5}{h{({\theta }_{q})}^{6}}\left[(1+\frac{\gamma }{2}\text{cos}(3{\theta }_{q}))\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}+\frac{\gamma \sqrt{\text{54}}}{6{q}_{\text{II}}^{2}}\text{dev}({t}_{\text{ij}})\right]\frac{\partial (h({\theta }_{q}))}{\partial {q}_{\text{mn}}}+\frac{1}{h{({\theta }_{q})}^{5}}(1+\frac{\gamma }{2}\text{cos}(3{\theta }_{q}))(\frac{{\delta }_{\text{im}}{\delta }_{\text{jn}}}{{q}_{\text{II}}}-\frac{{q}_{\text{ij}}{q}_{\text{mn}}}{{q}_{\text{II}}^{3}})\\ +\frac{1}{h{({\theta }_{q})}^{5}}\frac{\gamma }{2}\frac{{q}_{\text{ij}}\sqrt{\text{54}}}{{q}_{\text{II}}^{4}}({t}_{\text{mn}}-3\frac{\text{det}q}{{q}_{\text{II}}^{2}}{q}_{\text{mn}})+\frac{1}{h{({\theta }_{q})}^{5}}\frac{\gamma }{6}\frac{\sqrt{\text{54}}}{{q}_{\text{II}}^{2}}(\frac{\partial {t}_{\text{ij}}^{d}}{\partial {q}_{\text{mn}}}-{\mathrm{2t}}_{\text{ij}}^{d}\frac{{q}_{\text{mn}}}{{q}_{\text{II}}^{2}})\end{array}\)

The expression for \(\frac{\partial {t}_{\text{ij}}^{d}}{\partial {q}_{\text{mn}}}\) is explained as follows:

\(\frac{\partial {t}^{d}}{\partial {q}_{\text{11}}}=\left[\begin{array}{c}-\frac{1}{3}({q}_{\text{22}}+{q}_{\text{33}})\\ \frac{1}{3}(-{q}_{\text{22}}+2{q}_{\text{33}})\\ \frac{1}{3}(2{q}_{\text{22}}-{q}_{\text{33}})\\ 0\\ 0\\ -{q}_{\text{23}}\end{array}\right]\), \(\frac{\partial {t}^{d}}{\partial {q}_{\text{22}}}=\left[\begin{array}{c}\frac{1}{3}(-{q}_{\text{11}}+2{q}_{\text{33}})\\ -\frac{1}{3}({q}_{\text{11}}+{q}_{\text{33}})\\ \frac{1}{3}(2{q}_{\text{11}}-{q}_{\text{33}})\\ 0\\ -{q}_{\text{13}}\\ 0\end{array}\right]\), \(\frac{\partial {t}^{d}}{\partial {q}_{\text{33}}}=\left[\begin{array}{c}\frac{1}{3}(-{q}_{\text{11}}+2{q}_{\text{22}})\\ \frac{1}{3}(2{q}_{\text{11}}-{q}_{\text{22}})\\ -\frac{1}{3}({q}_{\text{11}}+{q}_{\text{22}})\\ -{q}_{\text{12}}\\ 0\\ 0\end{array}\right]\),

\(\frac{\partial {t}^{d}}{\partial {q}_{\text{12}}}=\left[\begin{array}{c}\frac{2}{3}{q}_{\text{12}}\\ \frac{2}{3}{q}_{\text{12}}\\ -\frac{4}{3}{q}_{\text{12}}\\ -{q}_{\text{33}}\\ {q}_{\text{23}}\\ {q}_{\text{13}}\end{array}\right]\), \(\frac{\partial {t}^{d}}{\partial {q}_{\text{13}}}=\left[\begin{array}{c}\frac{2}{3}{q}_{\text{13}}\\ -\frac{4}{3}{q}_{\text{13}}\\ \frac{2}{3}{q}_{\text{13}}\\ {q}_{\text{23}}\\ -{q}_{\text{22}}\\ {q}_{\text{12}}\end{array}\right]\), \(\frac{\partial {t}^{d}}{\partial {q}_{\text{23}}}=\left[\begin{array}{c}-\frac{4}{3}{q}_{\text{23}}\\ \frac{2}{3}{q}_{\text{23}}\\ \frac{2}{3}{q}_{\text{23}}\\ {q}_{\text{13}}\\ {q}_{\text{12}}\\ -{q}_{\text{11}}\end{array}\right]\)

  • calculation of \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

We have:

\(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}-(\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}{n}_{\text{rs}})\frac{\partial {n}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}-(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}{n}_{\text{rs}}+\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}\frac{\partial {n}_{\text{rs}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}){n}_{\text{mn}}\)

Define tensor \(\tilde{n}\) by \({\tilde{n}}_{\text{ij}}={\mathrm{\beta }}^{\text{'}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}+{\mathrm{\delta }}_{\text{ij}}\)

That is to say that \(n\) is then given by \({n}_{\text{ij}}=\frac{{\tilde{n}}_{\text{ij}}}{{\tilde{n}}_{\text{II}}}\) with \({\tilde{n}}_{\text{II}}=\sqrt{{\mathrm{\beta }}^{\text{'}2}+3}\)

In practice, to calculate \({\mathrm{\beta }}^{\text{'}}\), we use \({\mathrm{\Delta \varepsilon }}_{\text{ij}}\) instead of \({\mathrm{\Delta \varepsilon }}_{\text{ij}}^{\text{dp}}\), that is to say we have:

\({\mathrm{\beta }}^{\text{'}}=\mathrm{\beta }(\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}}-1)\text{signe}({s}_{\text{ij}}{\mathrm{\Delta \varepsilon }}_{\text{ij}})\)

So we have for \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}-(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}{n}_{\text{rs}}){n}_{\text{mn}}-(\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}\frac{\partial {\tilde{n}}_{\text{rs}}}{\partial {\mathrm{\sigma }}_{\text{kl}}})\frac{{\tilde{n}}_{\text{mn}}}{{\tilde{n}}_{\text{II}}^{2}}-(\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}{\tilde{n}}_{\text{rs}})\frac{\partial {\tilde{n}}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\frac{1}{{\tilde{n}}_{\text{II}}^{2}}-(\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}{\tilde{n}}_{\text{rs}}){\tilde{n}}_{\text{mn}}\frac{\partial (\frac{1}{{\tilde{n}}_{\text{II}}^{2}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

with:

\(\frac{\partial (\frac{1}{{\tilde{{n}^{2}}}_{\text{II}}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{\partial (\frac{1}{({\mathrm{\beta }}^{\text{'}2}+3)})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-\frac{1}{{({\mathrm{\beta }}^{\text{'}2}+3)}^{2}}\frac{\partial ({\mathrm{\beta }}^{\text{'}2})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-\frac{2{\mathrm{\beta }}^{2}(\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}}-1)}{{({\mathrm{\beta }}^{\text{'}2}+3)}^{2}}\frac{\partial (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

  • calculation of \(\frac{\partial (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial (\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{c}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{1}{{s}_{\text{II}}^{c}}\frac{\partial ({s}_{\text{II}})}{\partial {\mathrm{\sigma }}_{\text{kl}}}-\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{{c}^{2}}}\frac{\partial ({s}_{\text{II}}^{c})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=\frac{1}{{s}_{\text{II}}^{c}}\frac{\partial ({s}_{\text{II}})}{\partial {s}_{\text{mn}}}\frac{\partial {s}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}-\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{{c}^{2}}}\frac{\partial (-\frac{{R}_{c}({I}_{1}+{Q}_{\text{init}})}{h({\mathrm{\theta }}_{s})})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=\frac{1}{{s}_{\text{II}}^{c}}\frac{{s}_{\text{mn}}}{{s}_{\text{II}}}({\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}-\frac{1}{3}{\mathrm{\delta }}_{\text{mn}}{\mathrm{\delta }}_{\text{kl}})-\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{{c}^{2}}}(-\frac{{R}_{c}}{h({\mathrm{\theta }}_{s})}\frac{\partial {I}_{1}}{\partial {\mathrm{\sigma }}_{\text{kl}}}+\frac{{R}_{c}({I}_{1}+{Q}_{\text{init}})}{h{({\mathrm{\theta }}_{s})}^{2}}\frac{\partial h({\mathrm{\theta }}_{s})}{\partial {\mathrm{\sigma }}_{\text{kl}}})\)

\(=\frac{1}{{s}_{\text{II}}^{c}}\frac{{s}_{\text{mn}}}{{s}_{\text{II}}}({\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}-\frac{1}{3}{\mathrm{\delta }}_{\text{mn}}{\mathrm{\delta }}_{\text{kl}})-\frac{{s}_{\text{II}}}{{s}_{\text{II}}^{{c}^{2}}}(-\frac{{R}_{c}}{h({\mathrm{\theta }}_{s})}{\mathrm{\delta }}_{\text{kl}}+\frac{{R}_{c}({I}_{1}+{Q}_{\text{init}})}{h{({\mathrm{\theta }}_{s})}^{2}}\frac{\partial h({\mathrm{\theta }}_{s})}{\partial {s}_{\text{rs}}}\frac{\partial {s}_{\text{rs}}}{\partial {\mathrm{\sigma }}_{\text{kl}}})\)

  • calculation of \(\frac{\partial {\tilde{n}}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial {\tilde{n}}_{\text{mn}}}{\partial {\sigma }_{\text{kl}}}=\beta (\frac{1}{{s}_{\text{II}}^{c}}-\frac{1}{{s}_{\text{II}}})\text{signe}({s}_{\text{ij}}\Delta {\varepsilon }_{\text{ij}})\frac{\partial {s}_{\text{mn}}}{\partial {\sigma }_{\text{kl}}}+\beta \text{signe}({s}_{\text{ij}}\Delta {\varepsilon }_{\text{ij}}){s}_{\text{mn}}(\frac{\partial (\frac{1}{{s}_{\text{II}}^{c}})}{\partial {\sigma }_{\text{kl}}}-\frac{\partial (\frac{1}{{s}_{\text{II}}})}{\partial {\sigma }_{\text{kl}}})\)

\(=\beta (\frac{1}{{s}_{\text{II}}^{c}}-\frac{1}{{s}_{\text{II}}})\text{signe}({s}_{\text{ij}}\Delta {\varepsilon }_{\text{ij}})({\delta }_{\text{mk}}{\delta }_{\text{nl}}-\frac{1}{3}{\delta }_{\text{mn}}{\delta }_{\text{kl}})+\beta \text{signe}({s}_{\text{ij}}\Delta {\varepsilon }_{\text{ij}}){s}_{\text{mn}}(\frac{1}{{s}_{\text{II}}^{2}}\frac{\partial ({s}_{\text{II}})}{\partial {\sigma }_{\text{kl}}}-\frac{1}{{s}_{\text{II}}^{{c}^{2}}}\frac{\partial {s}_{\text{II}}^{c}}{\partial {\sigma }_{\text{kl}}})\)

  • calculation of \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial R}\):

\(\frac{\partial {G}_{\text{mn}}^{d}}{\partial R}=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial R}-(\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}{n}_{\text{rs}})\frac{\partial {n}_{\text{mn}}}{\partial R}-(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}})}{\partial R}{n}_{\text{rs}}+\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}\frac{\partial {n}_{\text{rs}}}{\partial R}){n}_{\text{mn}}\)

\(=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial R}-(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}})}{\partial R}{n}_{\text{rs}}){n}_{\text{mn}}\)

\(={\mathrm{\delta }}_{\text{mn}}-({\mathrm{\delta }}_{\text{rs}}{n}_{\text{rs}}){n}_{\text{mn}}\)

\(=\frac{{\mathrm{\beta }}^{\text{'}2}{\mathrm{\delta }}_{\text{mn}}-3{\mathrm{\beta }}^{\text{'}}\frac{{s}_{\text{mn}}}{{s}_{\text{II}}}}{{\mathrm{\beta }}^{\text{'}2}+3}\)

  • calculation of \(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {X}_{\text{kl}}}\):

\(\frac{\partial {G}_{\text{mn}}^{d}}{\partial {X}_{\text{kl}}}=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial {X}_{\text{kl}}}-(\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}{n}_{\text{rs}})\frac{\partial {n}_{\text{mn}}}{\partial {X}_{\text{kl}}}-(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}})}{\partial {X}_{\text{kl}}}{n}_{\text{rs}}+\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}}\frac{\partial {n}_{\text{rs}}}{\partial {X}_{\text{kl}}}){n}_{\text{mn}}\)

\(=\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial {X}_{\text{kl}}}-(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{rs}}})}{\partial {X}_{\text{kl}}}{n}_{\text{rs}}){n}_{\text{mn}}\)

  • calculation of \(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial {X}_{\text{kl}}}\):

\(\frac{\partial (\frac{\partial {f}^{d}}{\partial {\mathrm{\sigma }}_{\text{mn}}})}{\partial {X}_{\text{kl}}}=\frac{\partial {Q}_{\text{mn}}}{\partial {X}_{\text{kl}}}-(\frac{\partial {Q}_{\text{rs}}}{\partial {X}_{\text{kl}}}{X}_{\text{rs}}+{Q}_{\text{rs}}\frac{\partial {X}_{\text{rs}}}{\partial {X}_{\text{kl}}}){\mathrm{\delta }}_{\text{mn}}\)

\(=\frac{\partial {Q}_{\text{mn}}}{\partial {q}_{\text{ij}}}\frac{\partial {q}_{\text{ij}}}{\partial {X}_{\text{kl}}}-((\frac{\partial {Q}_{\text{rs}}}{\partial {q}_{\text{ij}}}\frac{\partial {q}_{\text{ij}}}{\partial {X}_{\text{kl}}}){X}_{\text{rs}}+{Q}_{\text{rs}}{\mathrm{\delta }}_{\text{kr}}{\mathrm{\delta }}_{\text{ls}}){\mathrm{\delta }}_{\text{mn}}\)

\(=-{I}_{1}\frac{\partial {Q}_{\text{mn}}}{\partial {q}_{\text{ij}}}{\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}}-((-{I}_{1}\frac{\partial {Q}_{\text{rs}}}{\partial {q}_{\text{ij}}}{\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}}){X}_{\text{rs}}+{Q}_{\text{rs}}{\mathrm{\delta }}_{\text{kr}}{\mathrm{\delta }}_{\text{ls}}){\mathrm{\delta }}_{\text{mn}}\)

  • calculation of \(\frac{\partial {G}_{\text{ij}}^{X}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial {G}_{\text{ij}}^{X}}{\partial {\sigma }_{\text{kl}}}=-\frac{1}{2b}({Q}_{\text{ij}}+\varphi {X}_{\text{ij}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\frac{\partial {I}_{1}}{\partial {\sigma }_{\text{kl}}}+\frac{1}{b}(\frac{\partial {Q}_{\text{ij}}}{\partial {\sigma }_{\text{kl}}}+\frac{\partial \varphi }{\partial {\sigma }_{\text{kl}}}{X}_{\text{ij}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\)

\(\begin{array}{}=-\frac{1}{2b}({Q}_{\text{ij}}+\varphi {X}_{\text{ij}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}{\delta }_{\text{kl}}+\frac{1}{b}\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}\frac{\partial {q}_{\text{mn}}}{\partial {\sigma }_{\text{kl}}}({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\\ +\frac{1}{b}(h({\theta }_{s}){Q}_{\text{II}}\frac{\partial {\varphi }_{o}}{\partial {\sigma }_{\text{kl}}}+{\varphi }_{o}{Q}_{\text{II}}\frac{\partial h({\theta }_{s})}{\partial {\sigma }_{\text{kl}}}+{\varphi }_{o}h({\theta }_{s})\frac{\partial {Q}_{\text{II}}}{\partial {\sigma }_{\text{kl}}}){X}_{\text{ij}}({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\end{array}\)

  • calculation of \(\frac{\partial h({\mathrm{\theta }}_{s})}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial h({\mathrm{\theta }}_{s})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=\frac{\partial h({\mathrm{\theta }}_{s})}{\partial {s}_{\text{mn}}}\frac{\partial {s}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=(\frac{\mathrm{\gamma }\sqrt{\text{54}}}{6h{({\mathrm{\theta }}_{s})}^{5}{q}_{\text{II}}^{3}}{t}_{\text{mn}}-\frac{\mathrm{\gamma }\text{cos}({\mathrm{3\theta }}_{q})}{\mathrm{2h}{({\mathrm{\theta }}_{s})}^{5}{q}_{\text{II}}^{2}}{s}_{\text{mn}})({\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}-\frac{1}{3}{\mathrm{\delta }}_{\text{mn}}{\mathrm{\delta }}_{\text{kl}})\)

  • calculation of \(\frac{\partial {Q}_{\text{II}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\frac{\partial {Q}_{\text{II}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=(\frac{\partial {Q}_{\text{II}}}{\partial {Q}_{\text{rs}}}\frac{\partial {Q}_{\text{rs}}}{\partial {q}_{\text{mn}}})\frac{\partial {q}_{\text{mn}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

\(=(\frac{{Q}_{\text{rs}}}{{Q}_{\text{II}}}\frac{\partial {Q}_{\text{rs}}}{\partial {q}_{\text{mn}}})({\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}-{\mathrm{\delta }}_{\text{mn}}(\frac{1}{3}{\mathrm{\delta }}_{\text{kl}}+{X}_{\text{kl}}))\)

  • calculation of \(\frac{\partial {\mathrm{\varphi }}_{o}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\):

\(\begin{array}{}\frac{\partial {\varphi }_{o}}{\partial {\sigma }_{\text{kl}}}=\frac{1}{{R}_{r}-\frac{h({\theta }_{s})}{h({\theta }_{q})}{R}_{m}\text{cos}({\theta }_{s}-{\theta }_{q})}\frac{\partial \text{cos}\alpha }{\partial {\sigma }_{\text{kl}}}\\ -\text{cos}\alpha \frac{\left[\frac{\partial {R}_{r}}{\partial {\sigma }_{\text{kl}}}-\frac{1}{h({\theta }_{q})}{R}_{m}\text{cos}({\theta }_{s}-{\theta }_{q})\frac{\partial h({\theta }_{s})}{\partial {\sigma }_{\text{kl}}}+\frac{h({\theta }_{s})}{h{({\theta }_{q})}^{2}}{R}_{m}\text{cos}({\theta }_{s}-{\theta }_{q})\frac{\partial h({\theta }_{s})}{\partial {\sigma }_{\text{kl}}}-\frac{h({\theta }_{s})}{h({\theta }_{q})}{R}_{m}\frac{\partial \text{cos}({\theta }_{s}-{\theta }_{q})}{\partial {\sigma }_{\text{kl}}}\right]}{{\left[{R}_{r}-\frac{h({\theta }_{s})}{h({\theta }_{q})}{R}_{m}\text{cos}({\theta }_{s}-{\theta }_{q})\right]}^{2}}\end{array}\)

with:

\(\begin{array}{}\frac{\partial \text{cos}\alpha }{\partial {\sigma }_{\text{kl}}}=\frac{1}{2{s}_{\text{II}}{I}_{1}{X}_{\text{II}}}(2{q}_{\text{II}}\frac{\partial {q}_{\text{II}}}{\partial {\sigma }_{\text{kl}}}-2{I}_{1}{X}_{\text{II}}^{2}\frac{\partial {I}_{1}}{\partial {\sigma }_{\text{kl}}}-2{s}_{\text{II}}\frac{\partial {s}_{\text{II}}}{\partial {\sigma }_{\text{kl}}})\\ -\frac{{q}_{\text{II}}^{2}-{({I}_{1}{X}_{\text{II}})}^{2}-{s}_{\text{II}}^{2}}{{s}_{\text{II}}{I}_{1}{X}_{\text{II}}}({s}_{\text{II}}{X}_{\text{II}}\frac{\partial {I}_{1}}{\partial {\sigma }_{\text{kl}}}+{I}_{1}{X}_{\text{II}}\frac{\partial {s}_{\text{II}}}{\partial {\sigma }_{\text{kl}}})\end{array}\)

\(=\frac{1}{{s}_{\text{II}}{I}_{1}{X}_{\text{II}}}\left[({q}_{\text{kl}}-{I}_{1}{X}_{\text{II}}^{2}{\mathrm{\delta }}_{\text{kl}}-{s}_{\text{kl}})-({q}_{\text{II}}^{2}-{({I}_{1}{X}_{\text{II}})}^{2}-{s}_{\text{II}}^{2})({s}_{\text{II}}{X}_{\text{II}}{\mathrm{\delta }}_{\text{kl}}+{I}_{1}{X}_{\text{II}}\frac{{s}_{\text{kl}}}{{s}_{\text{II}}})\right]\)

\(\frac{\partial {R}_{r}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-\frac{\mathrm{\mu }}{{I}_{1}+{Q}_{\text{init}}}{\mathrm{\delta }}_{\text{kl}}\)

\(\frac{\partial \text{cos}({\mathrm{\theta }}_{s}-{\mathrm{\theta }}_{q})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-\text{sin}({\mathrm{\theta }}_{s}-{\mathrm{\theta }}_{q})(\frac{\partial {\mathrm{\theta }}_{s}}{\partial {\mathrm{\sigma }}_{\text{kl}}}-\frac{\partial {\mathrm{\theta }}_{q}}{\partial {\mathrm{\sigma }}_{\text{kl}}})\)

  • calculation of \(\frac{\partial {G}_{\text{ij}}^{X}}{\partial {X}_{\text{kl}}}\) :

\(\frac{\partial {G}_{\text{ij}}^{X}}{\partial {X}_{\text{kl}}}=\frac{1}{b}(\frac{\partial {Q}_{\text{ij}}}{\partial {X}_{\text{kl}}}+\mathrm{\varphi }\frac{\partial {X}_{\text{ij}}}{\partial {X}_{\text{kl}}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\)

\(=\frac{1}{b}(\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}\frac{\partial {q}_{\text{mn}}}{\partial {X}_{\text{kl}}}+\mathrm{\varphi }{\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\)

\(=\frac{1}{b}(-{I}_{1}\frac{\partial {Q}_{\text{ij}}}{\partial {q}_{\text{mn}}}{\mathrm{\delta }}_{\text{mk}}{\mathrm{\delta }}_{\text{nl}}+\mathrm{\varphi }{\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}})({I}_{1}+{Q}_{\text{init}}){(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{-\mathrm{1,5}}\)

4.4.3. Convergence test#

The convergence criterion remains \(\frac{\parallel {\text{DY}}^{\text{p+1}}\parallel }{\parallel {Y}^{\text{p+1}}-{Y}^{0}\parallel }\) RESI_INTE_RELA.

4.5. Integration of nonlinear elastic, isotropic plastic and deviatory plastic mechanisms#

In this case, the new constraint state \({\mathrm{\sigma }}^{+}\) checks:

\({\mathrm{\sigma }}_{\text{ij}}^{+}={\mathrm{\sigma }}_{\text{ij}}^{-}+{D}_{\text{ijkl}}({\mathrm{\sigma }}^{+})({\mathrm{\Delta \varepsilon }}_{\text{kl}}-{\mathrm{\Delta \varepsilon }}_{\text{kl}}^{\text{ip}}-{\mathrm{\Delta \varepsilon }}_{\text{kl}}^{\text{dp}})\)

Taking into account the above, it can be deduced that the nonlinear system to be solved is composed of:

  • \({\text{LE}}_{\text{ij}}\): the elastic state law: \({\mathrm{\sigma }}_{\text{ij}}^{+}-{\mathrm{\sigma }}_{\text{ij}}^{-}-{D}_{\text{ijkl}}({\mathrm{\sigma }}^{+})({\mathrm{\Delta \varepsilon }}_{\text{kl}}+\frac{1}{3}{\mathrm{\Delta \lambda }}^{i}{\mathrm{\delta }}_{\text{kl}}-{\mathrm{\Delta \lambda }}^{d}{G}_{\text{kl}}^{d}({\mathrm{\sigma }}^{+},{R}^{+},{X}^{+}))=0\)

  • \(\text{LQ}\): the law of hardening of the internal variable \({Q}_{\text{iso}}\): \({Q}_{\text{iso}}^{+}-{Q}_{\text{iso}}^{-}-{\mathrm{D\lambda }}^{i}{G}^{{Q}_{\text{iso}}}({Q}_{\text{iso}}^{+})=0\)

  • \(\text{LR}\): the law of hardening of the variable \(R\): \({R}^{+}-{R}^{-}-{\mathrm{\Delta \lambda }}^{d}{G}^{R}({\mathrm{\sigma }}^{+},{R}^{+})=0\)

  • \({\text{LX}}_{\text{ij}}\): the law of work-hardening of the variable \({X}_{\text{ij}}\): \({X}_{\text{ij}}^{+}-{X}_{\text{ij}}^{-}-{\mathrm{\Delta \lambda }}^{d}{G}_{\text{ij}}^{X}({\mathrm{\sigma }}^{+},{X}^{+})=0\)

  • \(\text{FI}\): the isotropic charge area equation: \(-\frac{{I}_{1}^{+}+{Q}_{\text{init}}}{3}+{Q}_{\text{iso}}^{+}=0\)

  • \(\text{FD}\): the equation of the deviatory load area: \({q}_{\text{II}}^{+}h({\mathrm{\theta }}_{q}^{+})+{R}^{+}({I}_{1}^{+}+{Q}_{\text{init}})=0\)

As in the preceding paragraphs, the system \(R(Y)=0\) is solved by Newton’s method, where the unknown \(Y\) is given by \(Y=({\mathrm{\sigma }}_{\text{ij}}^{+},{Q}_{\text{iso}}^{+},{R}^{+},{X}_{\text{ij}}^{+},{\mathrm{\Delta \lambda }}^{i},{\mathrm{\Delta \lambda }}^{d})\) and where \(R=({\text{LE}}_{\text{ij}},\text{LQ},\text{LR},{\text{LX}}_{\text{ij}},\text{FI},\text{FD})\).

4.5.1. Initialization and test solution#

Starting with the state at the moment \(t\) \(({\mathrm{\sigma }}_{\text{ij}}^{-},{Q}_{\text{iso}}^{-},{R}^{-},{X}_{\text{ij}}^{-})\), we are looking for a test solution that brings us closer to the final solution. To do this we solve the following system of equations:

\(\{\begin{array}{c}{f}^{i}({s}_{\text{ij}}^{-}+{D}_{\text{ijkl}}^{+}({\mathrm{De}}_{\text{kl}}+\frac{1}{3}D{\lambda }^{i}{d}_{\text{kl}}-D{\lambda }^{d}{G}_{\text{kl}}^{d}),{Q}_{\text{iso}}^{-}+D{\lambda }^{i}{G}^{{Q}_{\text{iso}}-})=0\\ {f}^{d}({s}_{\text{ij}}^{-}+{D}_{\text{ijkl}}^{+}({\mathrm{De}}_{\text{kl}}+\frac{1}{3}D{\lambda }^{i}{d}_{\text{kl}}-D{\lambda }^{d}{G}_{\text{kl}}^{d}),{R}^{-}+D{\lambda }^{d}{G}^{R-},{X}_{\text{ij}}^{-}+D{\lambda }^{d}{G}_{\text{ij}}^{X-})=0\end{array}\)

with:

\({D}_{\text{ijkl}}^{-}={D}_{\text{ijkl}}({\mathrm{\sigma }}^{-})\), \({G}_{\text{kl}}^{d-}={G}_{\text{kl}}^{d}({\mathrm{\sigma }}^{-},{R}^{-},{X}^{-})\), \({G}^{{Q}_{\text{iso}}-}={G}^{{Q}_{\text{iso}}}({Q}_{\text{iso}}^{-})\) \({G}^{R-}={G}^{R}({\mathrm{\sigma }}^{-},{R}^{-})\), \({G}_{\text{ij}}^{X-}={G}_{\text{ij}}^{X}({\mathrm{\sigma }}^{-},{X}^{-})\) and where the unknowns are the plastic multipliers \(\Delta {\lambda }^{i}\) and \(\Delta {\lambda }^{d}\), by a single Newton’s iteration, that is to say finally that we have:

\(\begin{array}{c}{\frac{\mathrm{\partial }{f}^{i}}{\mathrm{\partial }\Delta {\lambda }^{i}}}_{\mathrm{\mid }\Delta {\lambda }^{i}\mathrm{=}\mathrm{0,}\Delta {\lambda }^{d}\mathrm{=}0}\Delta {\lambda }^{i}+{\frac{\mathrm{\partial }{f}^{i}}{\mathrm{\partial }\Delta {\lambda }^{d}}}_{\mathrm{\mid }\Delta {\lambda }^{i}\mathrm{=}\mathrm{0,}\Delta {\lambda }^{d}\mathrm{=}0}\Delta {\lambda }^{d}\mathrm{=}\mathrm{-}{f}^{{i}_{\mathrm{\mid }\Delta {\lambda }^{i}\mathrm{=}\mathrm{0,}\Delta {\lambda }^{d}\mathrm{=}0}}\\ {\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }\Delta {\lambda }^{i}}}_{\mathrm{\mid }\Delta {\lambda }^{i}\mathrm{=}\mathrm{0,}\Delta {\lambda }^{d}\mathrm{=}0}\Delta {\lambda }^{i}+{\frac{\mathrm{\partial }{f}^{d}}{\mathrm{\partial }\Delta {\lambda }^{d}}}_{\mathrm{\mid }\Delta {\lambda }^{i}\mathrm{=}\mathrm{0,}\Delta {\lambda }^{d}\mathrm{=}0}\Delta {\lambda }^{d}\mathrm{=}\mathrm{-}{f}^{{d}_{\mathrm{\mid }\Delta {\lambda }^{i}\mathrm{=}\mathrm{0,}\Delta {\lambda }^{d}\mathrm{=}0}}\end{array}\)

or again:

\({\mathrm{\Delta \lambda }}^{i}=\frac{\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{d}}{f}^{d}-\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{d}}{f}^{i}}{\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{i}}\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{d}}-\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{d}}\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{i}}}\) and \({\mathrm{\Delta \lambda }}^{d}=\frac{\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{i}}{f}^{i}-\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{i}}{f}^{d}}{\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{i}}\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{d}}-\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{d}}\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{i}}}\)

with:

\(\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{i}}=-({K}^{-}+{K}^{p-})\)

\(\frac{\partial {f}^{i}}{\partial {\mathrm{\Delta \lambda }}^{d}}={K}^{-}\text{tr}({G}^{d-})\)

\(\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{i}}=h({\mathrm{\theta }}_{q})\frac{\partial {q}_{\text{II}}}{\partial {\mathrm{\Delta \lambda }}^{i}}+{q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {\mathrm{\Delta \lambda }}^{i}}+({I}_{1}+{Q}_{\text{init}})\frac{\partial R}{\partial {\mathrm{\Delta \lambda }}^{i}}+R\frac{\partial {I}_{1}}{\partial {\mathrm{\Delta \lambda }}^{i}}\)

\(\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{d}}=h({\mathrm{\theta }}_{q})\frac{\partial {q}_{\text{II}}}{\partial {\mathrm{\Delta \lambda }}^{d}}+{q}_{\text{II}}\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {\mathrm{\Delta \lambda }}^{d}}+({I}_{1}+{Q}_{\text{init}})\frac{\partial R}{\partial {\mathrm{\Delta \lambda }}^{d}}+R\frac{\partial {I}_{1}}{\partial {\mathrm{\Delta \lambda }}^{d}}\)

We know that \(\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{d}}\) is calculated in the same way as before when only the plastic deviatory mechanism was activated. Moreover, for the calculation of \(\frac{\partial {f}^{d}}{\partial {\mathrm{\Delta \lambda }}^{i}}\) and when \({\mathrm{\Delta \lambda }}^{i}=0\) and \({\mathrm{\Delta \lambda }}^{d}=0\), we have the following relationships:

\(\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{i}}=\frac{1}{3}{D}_{\text{ijkl}}^{-}{\mathrm{\delta }}_{\text{kl}}-3{K}^{e-}(\frac{1}{3}{\mathrm{\delta }}_{\text{ij}}+{X}_{\text{ij}}^{-})\)

\(\frac{\partial {q}_{\text{II}}}{\partial {\mathrm{\Delta \lambda }}^{i}}=\frac{\partial {q}_{\text{II}}}{\partial {q}_{\text{ij}}}\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{i}}=\frac{{q}_{\text{ij}}}{{q}_{\text{II}}}(\frac{1}{3}{D}_{\text{ijkl}}^{-}{\mathrm{\delta }}_{\text{kl}}-3{K}^{e-}(\frac{1}{3}{\mathrm{\delta }}_{\text{ij}}+{X}_{\text{ij}}^{-}))\)

\(\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {\mathrm{\Delta \lambda }}^{i}}=\frac{\partial h({\mathrm{\theta }}_{q})}{\partial {q}_{\text{ij}}}\frac{\partial {q}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{i}}\)

\(\frac{\partial R}{\partial {\mathrm{\Delta \lambda }}^{i}}=0\)

\(\frac{\partial {I}_{1}}{\partial {\mathrm{\Delta \lambda }}^{i}}=3{K}^{-}\)

In short, we take \({Y}^{0}=({\mathrm{\sigma }}_{\text{ij}}^{0},{Q}_{\text{iso}}^{0},{R}^{0},{X}_{\text{ij}}^{0},{\mathrm{\Delta \lambda }}^{\mathrm{i0}},{\mathrm{\Delta \lambda }}^{\mathrm{d0}})\) for the test solution, with the following values:

\({\mathrm{\Delta \lambda }}^{\mathrm{i0}}\): the value found according to the previous formulation.

\({\mathrm{\Delta \lambda }}^{\mathrm{d0}}\): the value found according to the previous formulation.

\({\mathrm{\sigma }}_{\text{ij}}^{0}={\mathrm{\sigma }}_{\text{ij}}^{-}+{D}_{\text{ijkl}}^{-}({\mathrm{\Delta \varepsilon }}_{\text{kl}}+\frac{1}{3}{\mathrm{\Delta \lambda }}^{\mathrm{i0}}{\mathrm{\delta }}_{\text{kl}}-{\mathrm{\Delta \lambda }}^{\mathrm{d0}}{G}_{\text{kl}}^{d-})\)

\({Q}_{\text{iso}}^{0}={Q}_{\text{iso}}^{-}+{\mathrm{\Delta \lambda }}^{\mathrm{i0}}{G}^{{Q}_{\text{iso}}-}\)

\({R}^{0}={R}^{-}+{\mathrm{\Delta \lambda }}^{\mathrm{d0}}{G}^{R-}\)

\({X}_{\text{ij}}^{0}={X}_{\text{ij}}^{-}+{\mathrm{\Delta \lambda }}^{\mathrm{d0}}{G}_{\text{ij}}^{X-}\)

4.5.2. Newton iterations#

\(\frac{\text{DR}}{\text{DY}}\) is here given by:

\(\frac{\text{DR}}{\text{DY}}=\left[\begin{array}{cccccc}\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {Q}_{\text{iso}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial R}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {X}_{\text{kl}}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{i}}& \frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial \text{LQ}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial \text{LQ}}{\partial {Q}_{\text{iso}}}& \frac{\partial \text{LQ}}{\partial R}& \frac{\partial \text{LQ}}{\partial {X}_{\text{kl}}}& \frac{\partial \text{LQ}}{\partial {\mathrm{\Delta \lambda }}^{i}}& \frac{\partial \text{LQ}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial \text{LR}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial \text{LR}}{\partial {Q}_{\text{iso}}}& \frac{\partial \text{LR}}{\partial R}& \frac{\partial \text{LR}}{\partial {X}_{\text{kl}}}& \frac{\partial \text{LR}}{\partial {\mathrm{\Delta \lambda }}^{i}}& \frac{\partial \text{LR}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {Q}_{\text{iso}}}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial R}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {X}_{\text{kl}}}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{i}}& \frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial \text{FI}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial \text{FI}}{\partial {Q}_{\text{iso}}}& \frac{\partial \text{FI}}{\partial R}& \frac{\partial \text{FI}}{\partial {X}_{\text{kl}}}& \frac{\partial \text{FI}}{\partial {\mathrm{\Delta \lambda }}^{i}}& \frac{\partial \text{FI}}{\partial {\mathrm{\Delta \lambda }}^{d}}\\ \frac{\partial \text{FD}}{\partial {\mathrm{\sigma }}_{\text{kl}}}& \frac{\partial \text{FD}}{\partial {Q}_{\text{iso}}}& \frac{\partial \text{FD}}{\partial R}& \frac{\partial \text{FD}}{\partial {X}_{\text{kl}}}& \frac{\partial \text{FD}}{\partial {\mathrm{\Delta \lambda }}^{i}}& \frac{\partial \text{FD}}{\partial {\mathrm{\Delta \lambda }}^{d}}\end{array}\right]\)

where the new terms are void:

\(\frac{\partial \text{LQ}}{\partial R}=0\), \(\frac{\partial \text{LQ}}{\partial {X}_{\text{kl}}}=0\), \(\frac{\partial \text{LQ}}{\partial {\mathrm{\Delta \lambda }}^{d}}=0\), \(\frac{\partial \text{LR}}{\partial {Q}_{\text{iso}}}=0\), \(\frac{\partial \text{LR}}{\partial {\mathrm{\Delta \lambda }}^{i}}=0\), \(\frac{\partial {\text{LX}}_{\text{ij}}}{\partial {Q}_{\text{iso}}}=0\),

\(\frac{\partial {\text{LX}}_{\text{ij}}}{\partial {\mathrm{\Delta \lambda }}^{i}}=0\), \(\frac{\partial \text{FI}}{\partial R}=0\), \(\frac{\partial \text{FI}}{\partial {X}_{\text{kl}}}=0\), \(\frac{\partial \text{FI}}{\partial {\mathrm{\Delta \lambda }}^{d}}=0\),, \(\frac{\partial \text{FD}}{\partial {Q}_{\text{iso}}}=0\), \(\frac{\partial \text{FD}}{\partial {\mathrm{\Delta \lambda }}^{i}}=0\)

and where the terms already defined remain unchanged, except for \(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\) which becomes:

\(\frac{\partial {\text{LE}}_{\text{ij}}}{\partial {\mathrm{\sigma }}_{\text{kl}}}={\mathrm{\delta }}_{\text{ik}}{\mathrm{\delta }}_{\text{jl}}-{D}_{\text{ijmn}}^{\text{lineaire}}({\mathrm{\Delta \varepsilon }}_{\text{mn}}+\frac{1}{3}{\mathrm{\Delta \lambda }}^{i}{\mathrm{\delta }}_{\text{mn}}-{\mathrm{\Delta \lambda }}^{d}{G}_{\text{mn}}^{d})\frac{n}{3{P}_{a}}{(\frac{{I}_{1}+{Q}_{\text{init}}}{3{P}_{a}})}^{n-1}{\mathrm{\delta }}_{\text{kl}}+{D}_{\text{ijmn}}{\mathrm{\Delta \lambda }}^{d}\frac{\partial {G}_{\text{mn}}^{d}}{\partial {\mathrm{\sigma }}_{\text{kl}}}\)

4.5.3. Convergence test#

The convergence criterion remains \(\frac{\parallel {\text{DY}}^{\text{p+1}}\parallel }{\parallel {Y}^{\text{p+1}}-{Y}^{0}\parallel }\) RESI_INTE_RELA

4.6. Relaxation procedure based on an estimate of the normals at the deviatory load surface#

When the deviatory plastic mechanism intervenes, a relaxation procedure within Newton’s iterations is taken into account. This makes it possible to avoid certain oscillation problems in the calculation of solution \({Y}^{p+1}\) which ultimately lead to the non-convergence of numerical integration.

So in iteration \(p+1\), instead of refreshing the unknown \({Y}^{p+1}\) by a full increment \({\mathrm{\deltay }}^{p+1}\)

\({Y}^{p+1}={Y}^{p}+\mathrm{\delta }{Y}^{p+1}\)

We pose

\({Y}_{m}^{p+1}={Y}^{p}+{\mathrm{\rho }}_{m}\mathrm{\delta }{Y}^{p+1}\)

and we try, by looping over sub-iterations \(m\), to determine an optimal value of the scalar \({\mathrm{\rho }}_{m}\). This value is sought by considering the rotation of the normal, in the deviatory plane, to surface \({f}^{d}\), during sub-iterations. This normal, noted \({\tilde{n}}_{m}\), is expressed using the constraints contained in the term \({Y}_{m}^{p+1}\) by

\({\tilde{n}}_{m}=2h{({\mathrm{\theta }}_{q})}^{5}{q}_{\text{II}}\frac{\partial {f}^{d}}{\partial {q}_{\text{ij}}}=(2+\mathrm{\gamma }\text{cos}(3{\mathrm{\theta }}_{q})){q}_{\text{ij}}+\frac{\sqrt{6}\mathrm{\gamma }}{{q}_{\text{II}}}\frac{\partial \text{det}(q)}{\partial {q}_{\text{ij}}}\)

Starting with the initial value \({\mathrm{\rho }}_{0}=1\text{.}0\), the process put in place consists of the following steps:

  • calculation of normals \({\tilde{n}}_{m-1}\) and \({\tilde{n}}_{m}\)

  • calculation of the rotation angle \({\phi }_{m}\) between these normals: \(\text{cos}{\mathrm{\phi }}_{m}=\frac{{\tilde{n}}_{m-1}:{\tilde{n}}_{m}}{\sqrt{{\tilde{n}}_{m-1}}\sqrt{{\tilde{n}}_{m}}}\)

  • evolution test \(\text{cos}{\mathrm{\phi }}_{m}\):

If \(\text{cos}{\mathrm{\phi }}_{m}\le \text{TOLROT}\) then \({\mathrm{\rho }}_{m+1}=\text{DECREL}{\mathrm{\rho }}_{m}\) and \(m=m+1\)

Otherwise end of sub-iterations and \({Y}^{p+1}={Y}_{m}^{p+1}\)

4.7. Redistribution of the time step#

As with most behavioral relationships, the possibility of locally (at Gauss points) the time step was introduced for the CJS model to facilitate numerical integration. This possibility is managed by the ITER_INTE_PAS operand of the CONVERGENCE keyword of the STAT_NON_LINE operator. If itepas, the value of ITER_INTE_PAS, is 0, 1, or -1 there is no redistribution (note: 0 is the default value). If itepas is positive the redistribution is automatic, if it is negative the redistribution is only taken into account in case of non-convergence with the initial time step.

Redistribution consists in integrating, after the elastic prediction phase, the integration of the plastic mechanism (s) involved with a deformation increment whose components correspond to the components of the initial deformation increment divided by the absolute value of itepas.

4.8. Various remarks#

4.8.1. Calculating the term \(\text{cos}({\mathrm{\theta }}_{s}-{\mathrm{\theta }}_{q})\)#

The term \(\text{cos}({\mathrm{\theta }}_{s}-{\mathrm{\theta }}_{q})\) appears in the expression for \({\mathrm{\varphi }}_{o}\). For its calculation, we adopted the same method as that used in ECL. That is, we determine angles \({\mathrm{\theta }}_{s}\) and \({\mathrm{\theta }}_{q}\) in the following way:

\({\mathrm{\theta }}_{s}=\frac{1}{3}\text{Arctan}(\frac{\sqrt{\text{1-}{\text{cos}}^{2}(3{\mathrm{\theta }}_{s})}}{\text{cos}(3{\mathrm{\theta }}_{s})})\) and \({\mathrm{\theta }}_{q}=\frac{1}{3}\text{Arctan}(\frac{\sqrt{\text{1-}{\text{cos}}^{2}({\mathrm{3\theta }}_{q})}}{\text{cos}({\mathrm{3\theta }}_{q})})\)

then we take the cosine of the difference.

These expressions for \({\mathrm{\theta }}_{s}\) and \({\mathrm{\theta }}_{q}\) are also used to calculate:

\(\frac{\partial \text{cos}({\mathrm{\theta }}_{s}-{\mathrm{\theta }}_{q})}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-\text{sin}({\mathrm{\theta }}_{s}-{\mathrm{\theta }}_{q})(\frac{\partial {\mathrm{\theta }}_{s}}{\partial {\mathrm{\sigma }}_{\text{kl}}}-\frac{\partial {\mathrm{\theta }}_{q}}{\partial {\mathrm{\sigma }}_{\text{kl}}})\)

with \(\frac{\partial {\mathrm{\theta }}_{s}}{\partial {\mathrm{\sigma }}_{\text{kl}}}=-\frac{1}{3}\sqrt{1-{\text{cos}}^{2}({\mathrm{3\theta }}_{s})}\frac{\sqrt{\text{54}}}{{q}_{\text{II}}^{3}}({t}_{\text{kl}}-3\frac{\text{det}(q)}{{q}_{\text{II}}^{2}}{q}_{\text{kl}})\)

4.8.2. Calculating \({R}_{r}\)#

The break radius introduced in model CJS3 is given by the formula

\({R}_{r}={R}_{c}+\mathrm{\mu }\text{ln}(\frac{3{p}_{c}}{{I}_{1}+{Q}_{\text{init}}})\)

In fact, when \(\frac{{I}_{1}+{Q}_{\text{init}}}{3}>{p}_{c}\), you should block \({R}_{r}\) at the value of \({R}_{c}\). The dilatence domain disappears and it is not accepted that \({R}_{r}\) can decrease below \({R}_{c}\). Consequently, instead of the preceding formulation, the following expression is introduced

\({R}_{r}={R}_{c}+\mathrm{\mu }\text{max}\left[\mathrm{0,}\text{ln}(\frac{3{p}_{c}}{{I}_{1}+{Q}_{\text{init}}})\right]\)

4.8.3. Traction#

Without cohesion, the area of traction that corresponds to positive stresses is unacceptable for soils. From the point of view of the integration of model CJS, when the state of the stresses tends to the top of the cone of the load surface, the numerical risk of tipping into this forbidden domain increases. However, when one projects oneself or when one makes a prediction at a point in this field, numerical calculation leads either to an erroneous result or to a fatal error. In fact, traction is numerically expressed by a positive value of \({I}_{1}\). This value then poses a problem when evaluating certain quantities such as \({(\frac{{I}_{1}^{+}+{Q}_{\text{init}}}{3{P}_{a}})}^{-1\text{.}5}\); moreover, from a theoretical point of view, it would generate a negative \({q}_{\text{II}}\) value according to the equation of the deviatory load area.

Such a phenomenon has been detected at several levels: in particular in elastic prediction with model CJS1, and in general in local Newton iterations involving the deviatory mechanism. The same answer was given in order to overcome this pathology: it is a question of virtually projecting the constraints in the elastic domain onto the hydrostatic axis by posing:

\(\begin{array}{}{\mathrm{\sigma }}_{\text{11}}={\mathrm{\sigma }}_{\text{22}}={\mathrm{\sigma }}_{\text{33}}=-1\text{kPa}\\ {\mathrm{\sigma }}_{\text{12}}={\mathrm{\sigma }}_{\text{13}}={\mathrm{\sigma }}_{\text{23}}=0\end{array}\)

The stress state is thus repositioned in the compression domain with little departure from the initial unacceptable prediction envisaged, and in the hope that structural considerations will allow the global calculation to converge.

In addition, the internal variables do not change and it is assumed that we have returned to the elastic domain.