8. Appendix: Expression of the Jacobian matrix

We recall the expression for the non-linear system of equations systeme_non_lineaire and the definition of its Jacobian matrix jacobienne:

\[\]

boldsymbol {r} =

begin {Bmatrix} boldsymbol {r} _1\ r_2\ r_2\ r_3end {Bmatrix} =begin {Bmatrix} -Deltaboldsymbol {epsilon} +Deltaboldsymbol {epsilon} ^e+Deltar_3end {Bmatrix} =end {Bmatrix} =begin {Bmatrix} -Deltaboldsymbol {epsilon} +Deltaboldsymbol {epsilon} ^e+^e + +Deltaboldsymbol {epsilon} ^e + +Deltaboldsymbol {epsilon} ^e +^e +Deltalambdacfrac {partial f_ {n+1}} {partialboldsymbol {sigma}}}\ -Deltaxi+Deltalambdacfrac {partial f_ {n+1}} {partial p_ {c, n+1}}\ f_ {n+1} /K=0end {Bmatrix},qquad

J =begin {Bmatrix} cfrac {partialboldsymbol {r} _1} {partialDeltaboldsymbol {epsilon} ^e} &cfrac {partialboldsymbol {r} _1} {partialdelta} _1} {partialdelta_x} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1} {partialboldsymbol {r} _1}partial r_2} {partialDeltaboldsymbol {boldsymbol {epsilon} ^e} &cfrac {partial r_2} {partialdeltaxi} &cfrac {partial r_2} {partialpartial r_2} {partialpartial r_2} {partialpartial r_2} &cfrac {partial r_xi} &cfrac {partial r_2} &cfrac {partial r_2} &cfrac {partial r_2} {partial r_2} {partialpartial r_2} {partialpartial r_2} {partialpartial r_2} {partialpartial r_2} {partialpartial r_2} {partial} &cfrac {partial r_3} {partialDeltaxi} &cfrac {partial r_3} {partialDeltalambda}end {Bmatrix}

label:

Jacobian_appendix

In MFront, this \(J\) matrix can be obtained by numerical disturbance or analytically, as is the case presented below. Its components are detailed in the case of an increment with plastic correction. For convenience, note:

• The flow directions of \(\Delta\boldsymbol{\epsilon}^p\) and \(\Delta\xi\):

\[\]

begin {align}

&boldsymbol {N} _ {n+1} =frac {partial f_ {n+1}} {partialboldsymbol {sigma} _ {n+1}} =frac {cfrac {3} {3} {2M^2}frac {2} {2M^2}boldsymbol {A} _ {d, n+1} +A_ {m, n+1}} =frac {cfrac {3} {2M^2} {2M^2}boldsymbol {A} _ {d, n+1} +A_ {m, n+1}}frac {boldsymbol {I}} {3}} {T_ {eq} (boldsymbol {A} _ {n+1})}\ &N_ {v, n+1} =frac {partial f_ {n+1}} {partial p_ {c, n+1}} =mathrm {tr}left (boldsymbol {N} _ {n+1} _ {n+1}right)qquadpartial f_ {n+1}right)qquadtext {with}qquadboldsymbol {A} =boldsymbol {sigma} +p_cboldright symbol {I},quad T_ {eq} (boldsymbol {A}) =sqrt {left (frac {A_ {eq}} {M}right) ^2+A_m^2} end {align} :label: derivee_utile_1

• The following two derivatives:

\[\]

begin {align}

&boldsymbol {N} _ {n+1,boldsymbol {A} _ {n+1}} =frac {partialboldsymbol {N} _ {n+1}} {partialboldsymbol {A}} _ {n+1}} =frac {cfrac {3} {2M^2}}mathbb {K}} +cfrac {mathbb {K} +cfrac {mathbb {J}} {3} -boldsymbol {N} _ {n+1} _ {n+1}otimesboldsymbol {N} _ {n+1}} {T_ {eq} (boldsymbol {A} _ {n+1})}\ &N_ {v, n+1,boldsymbol {A} _ {n+1}} =frac {partial N_ {v, n+1}} {partialboldsymbol {A} _ {n+1}}} =frac {cfrac {cfrac {cfrac {boldsymbol {I}}} {3} -N_ {v, n+1}boldsymbol {N} _ {n+1}} =frac {cfrac {cfrac {boldsymbol {I}} {3} -N_ {v, n+1}}boldsymbol {N} _ {n+1}} =frac {cfrac {cfrac {boldsymbol {I}}} {3} -N_ {v, n+1}}} {T_ {eq} (boldsymbol {A} _ {n+1})} end {align} :label: derivee_utile_2

8.1. First line

The derivation of each term in the first line of the system shown jacobienne_annexe provides:

\[\]

begin {align}

&frac {partialboldsymbol {r} _1} {partialDeltaboldsymbol {epsilon} ^e} =mathbb {I} +Deltalambdaboldsymbol {N}_ {n} _ {n} _ {n+1} _ {n+1}}:mathbb {C} _ {n+1}\ &frac {partialboldsymbol {r} _1} {partialDeltalambda} =boldsymbol {N} _ {n+1}\ &frac {partialboldsymbol {r} _1} {partialDeltaxi} =Deltalambdaboldsymbol {N} _ {n+1,boldsymbol {A} _ {n+1}}}:frac {partialboldsymbol {A} _ {n+1}} {partialDeltaxi}} {partialDeltaxi} end {align} :label: jacobienne_line1

where \(\frac{\partial \boldsymbol{A}_{n+1}}{\partial \Delta\xi} = p_{c,n+1}'\boldsymbol{I} = -\beta p_{c,n+1}\boldsymbol{I}\) in accordance with the expression of critical pressure defined expression_pression_critique and \(\mathbb{I}\) denote the fourth-order identity tensor.

8.2. Second line

The second line of the system reads:

\[\]

begin {align}

&frac {partial r_2} {partialDeltaDeltaboldsymbol {epsilon} ^e} =Deltalambda N_ {v, n+1,boldsymbol {A} _ {n+1}}}:mathbb {C} _ {n+1}\ &frac {partial r_2} {partialDeltalambda} =N_ {v, n+1}\ &frac {partial r_2} {partialDeltaxi} =-1 +deltalambda N_ {v, n+1,boldsymbol {A} _ {n+1}}:frac {partialdeltadeltaxi}}:frac {partialdeltax+1}}:frac {partialdeltan+1}}:frac {partialdeltan+1}}:frac {partialdeltan+1}} end {align} :label: jacobienne_line2

8.3. Third line

Finally, the third line of the system is:

\[\]

begin {align}

&frac {partial r_3} {partialDeltaDeltaboldsymbol {epsilon} ^e} =boldsymbol {N} _ {n+1}:mathbb {C} _ {n+1} /K\ &frac {partial r_3} {partialDeltalambda} =-H_n/K\ &frac {partial r_3} {partialDeltaxi} =boldsymbol {N} _ {n+1}:frac {partialboldsymbol {A} _ {n+1}}} {partialDeltaxi} /K end {align} :label: jacobienne_line3