Appendix: Expression of the Jacobian matrix
============================================

.. _target to appendix:

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

.. math::
\ boldsymbol {r} =
 \ begin {Bmatrix}
 \ boldsymbol {r} _1\\ r_2\\ r_2\\ r_3\ end {Bmatrix} =\ begin {Bmatrix} -\ Delta\ boldsymbol {\ epsilon} +\ Delta\ boldsymbol {\ epsilon} ^e+\ Delta\ r_3\ end {Bmatrix} =\ end {Bmatrix} =\ begin {Bmatrix} -\ Delta\ boldsymbol {\ epsilon} +\ Delta\ boldsymbol {\ epsilon} ^e+\ ^e + +\ Delta\ boldsymbol {\ epsilon} ^e + +\ Delta\ boldsymbol {\ epsilon} ^e +\ ^e +\ Delta\ lambda\ cfrac {\ partial f_ {n+1}} {\ partial\ boldsymbol {\ sigma}}}\\ -\ Delta\ xi+\ Delta\ lambda\ cfrac {\ partial f_ {n+1}} {\ partial p_ {c, n+1}}\\ f_ {n+1} /K=0\ end {Bmatrix},\ qquad
  J =\ begin {Bmatrix}
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ epsilon} ^e} &\ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ delta} _1} {\ partial\ delta\ _x} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1} {\ partial\ boldsymbol {r} _1}\ partial r_2} {\ partial\ Delta\ boldsymbol {\ boldsymbol {\ epsilon} ^e} &\ cfrac {\ partial r_2} {\ partial\ delta\ xi} &\ cfrac {\ partial r_2} {\ partial\ partial r_2} {\ partial\ partial r_2} {\ partial\ partial r_2} &\ cfrac {\ partial r_xi} &\ cfrac {\ partial r_2} &\ cfrac {\ partial r_2} &\ cfrac {\ partial r_2} {\ partial r_2} {\ partial\ partial r_2} {\ partial\ partial r_2} {\ partial\ partial r_2} {\ partial\ partial r_2} {\ partial\ partial r_2} {\ partial} &\ cfrac {\ partial r_3} {\ partial\ Delta\ xi} &\ cfrac {\ partial r_3} {\ partial\ Delta\ lambda}\ end {Bmatrix}
 :label: Jacobian_appendix
 
In MFront, this :math:`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 :math:`\Delta\boldsymbol{\epsilon}^p` and :math:`\Delta\xi`:
 
.. math::
\ begin {align}
  &\ boldsymbol {N} _ {n+1} =\ frac {\ partial f_ {n+1}} {\ partial\ boldsymbol {\ 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)\ qquad\ partial f_ {n+1}\ right)\ qquad\ text {with}\ qquad\ boldsymbol {A} =\ boldsymbol {\ sigma} +p_c\ boldright 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:
 
.. math::
\ begin {align}  
  &\ boldsymbol {N} _ {n+1,\ boldsymbol {A} _ {n+1}} =\ frac {\ partial\ boldsymbol {N} _ {n+1}} {\ partial\ boldsymbol {A}} _ {n+1}} =\ frac {\ cfrac {3} {2M^2}}\ mathbb {K}} +\ cfrac {\ mathbb {K} +\ cfrac {\ mathbb {J}} {3} -\ boldsymbol {N} _ {n+1} _ {n+1}\ otimes\ boldsymbol {N} _ {n+1}} {T_ {eq} (\ boldsymbol {A} _ {n+1})}\\  
  &N_ {v, n+1,\ boldsymbol {A} _ {n+1}} =\ frac {\ partial N_ {v, n+1}} {\ partial\ boldsymbol {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
 
First line
--------------

The derivation of each term in the first line of the system shown :eq:`jacobienne_annexe` provides:
 
.. math::
\ begin {align}
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ epsilon} ^e} =\ mathbb {I} +\ Delta\ lambda\ boldsymbol {N}\ _ {n} _ {n} _ {n+1} _ {n+1}}:\ mathbb {C} _ {n+1}\\
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda} =\ boldsymbol {N} _ {n+1}\\
 &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ xi} =\ Delta\ lambda\ boldsymbol {N} _ {n+1,\ boldsymbol {A} _ {n+1}}}:\ frac {\ partial\ boldsymbol {A} _ {n+1}} {\ partial\ Delta\ xi}} {\ partial\ Delta\ xi} 
 \ end {align}
 :label: jacobienne_line1

where :math:`\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 :eq:`expression_pression_critique` and :math:`\mathbb{I}` denote the fourth-order identity tensor.

Second line
--------------

The second line of the system reads:

.. math::
\ begin {align}
 &\ frac {\ partial r_2} {\ partial\ Delta\ Delta\ boldsymbol {\ epsilon} ^e} =\ Delta\ lambda N_ {v, n+1,\ boldsymbol {A} _ {n+1}}}:\ mathbb {C} _ {n+1}\\
 &\ frac {\ partial r_2} {\ partial\ Delta\ lambda} =N_ {v, n+1}\\
 &\ frac {\ partial r_2} {\ partial\ Delta\ xi} =-1 +\ delta\ lambda N_ {v, n+1,\ boldsymbol {A} _ {n+1}}:\ frac {\ partial\ delta\ delta\ xi}}:\ frac {\ partial\ delta\ x+1}}:\ frac {\ partial\ delta\ n+1}}:\ frac {\ partial\ delta\ n+1}}:\ frac {\ partial\ delta\ n+1}}
 \ end {align}
 :label: jacobienne_line2
 
Third line
---------------

Finally, the third line of the system is:

.. math::
\ begin {align}
 &\ frac {\ partial r_3} {\ partial\ Delta\ Delta\ boldsymbol {\ epsilon} ^e} =\ boldsymbol {N} _ {n+1}:\ mathbb {C} _ {n+1} /K\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ lambda} =-H_n/K\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ xi} =\ boldsymbol {N} _ {n+1}:\ frac {\ partial\ boldsymbol {A} _ {n+1}}} {\ partial\ Delta\ xi} /K
 \ end {align}
 :label: jacobienne_line3