.. _r7.01.17-jacobien:

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

In the case where the plasticity criterion :math:`f_1` is not verified after the elastic prediction, we recall the expression of the system 
of nonlinear equations :eq:`residu_plastique` and the definition of the Jacobian matrix :eq:`jacobienne`:

.. math::
\ begin {Bmatrix}\ boldsymbol {r} _1\\ r_2\\ r_3\ end {Bmatrix} =
  \ begin {Bmatrix} 
  \ Delta\ boldsymbol {\ varepsilon} ^e -\ Delta\ boldsymbol {\ varepsilon} +\ Delta\ lambda_1\ cfrac {\ partial g_1} {\ partial\ boldsymbol {\ sigma} "}\ Bigg {\ sigma}"}\ Bigg {|} _ {n+1} -\ Delta\ lambda_1\ cfrac {\ partial g_1}} {\ partial\ boldsymbol {\ sigma} "}"}\ Bigg {|} _ {n+1} -\ Delta\ lambda_2\ cfrac {\ boldsymbol {I}} {3}\\
  \ cfrac {f_ {1, n+1}} {\ mu}\\
  \ Delta\ xi_1-\ Delta\ lambda_1\ boldsymbol {I}:\ cfrac {\ partial g_1} {\ partial\ boldsymbol {\ sigma} "}\ Bigg {|}\ Bigg {|} |} _ {n+1}\\
  \ end {Bmatrix},\ quad 
  J =\ begin {Bmatrix}
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda_1}} &
  \ cfrac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ xi_1}\\
  \ cfrac {\ partial r_2} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial r_2} {\ partial\ Delta\ lambda_1}} &
  \ cfrac {\ partial r_2} {\ partial\ Delta\ xi_1}\\
  \ cfrac {\ partial r_3} {\ partial\ Delta\ Delta\ boldsymbol {\ varepsilon} ^e}} &
  \ cfrac {\ partial r_3} {\ partial\ Delta\ lambda_1}} &
  \ cfrac {\ partial r_3} {\ partial\ Delta\ xi_1}
  \ end {Bmatrix}
  :label: Jacobienne_plastic_residure_appendix
 
In MFront, the :math:`J` matrix can be obtained by numerical disturbance or analytically, as is the case presented below. Its components are detailed below,
where we note for brevity:

* The derivative of the plasticity criterion with respect to the net stress tensor:

.. math::
\ boldsymbol {M} _ {n+1} =\ frac {\ partial f_1} {\ partial\ boldsymbol {\ sigma} "}\ Bigg {|} _ {n+1} =\ frac {\ cfrac {3} {3} {2} {2M^2}\ boldsymbol {A} _ {d, n+1} +A_ {m, n+1} =\ frac {\ n+1} =\ frac {\ n+1} =\ frac {\ boldsymbol {I}} {3}} {T_ {eq} {T_ {eq} (\ boldsymbol {A} _ {n+1})})},
 \ quad\ text {with}\ quad 
 \ boldsymbol {A} =\ boldsymbol {\ sigma} "+\ frac {\ tilde p_ {con} (p_c,\ varepsilon_v^p) -k_sp_c} {2}\ boldsymbol {I}" +\ frac {I} "+\ frac {I}" +\ frac {I} ",\ quad T_ {eq} (\ boldsymbol {A}) =\ sqrt {\ left (\ frac {A_) {eq}} {M}\ right) ^2+a_M^2}
 :label: derivee_utile_1
 
* The flow directions of :math:`\Delta\boldsymbol{\varepsilon}_1^p` and :math:`\Delta\xi_1`:
 
.. math::
\ begin {align}
  &\ boldsymbol {N} _ {n+1} =\ frac {\ cfrac {\ partial g_1} {\ partial\ boldsymbol {\ sigma} "}\ Bigg {|} _ {n+1} =\ frac {\ cfrac {\ cfrac {3\ frac {3\ alpha} {3\ alpha} {2M^2} {2M^2}\ boldsymbol {A} _ {d, n+1} +A_ {m, n+1} =\ frac {\ cfrac {\ cfrac {3\ alpha} {2M^2}\ cfrac {\ boldsymbol {I}} {3}} {3}} {T_ {eq,\ alpha} (\ boldsymbol {A} _ {n+1})})},
  \ quad\ text {with}\ quad
  T_ {eq,\ alpha} (\ boldsymbol {A}) =\ sqrt {\ alpha\ left (\ frac {A_ {eq}} {M}\ right) ^2+a_M^2}\\
  &N_ {v, n+1} =\ boldsymbol {I}:\ boldsymbol {N} _ {n+1}
  \ end {align}
  :label: derivee_utile_2
  
* Their following two derivatives:
 
.. math::
\ begin {align}  
  &\ boldsymbol {N} _ {n+1,\ boldsymbol {A} _ {n+1}} =\ frac {\ partial\ boldsymbol {N}} {\ partial\ boldsymbol {A}}}\ Bigg {|}}}\ Bigg {|} _ {n+1} =\ frac {\ cfrac {3\ partial\ boldsymbol {N}}} {\ partial\ boldsymbol {A}}}\ Bigg {| |}}}\ Bigg {| |} _ {n+1} =\ frac {\ cfrac {3\ alpha} {2M^2}}\ mathbb {K}}}\ bigg {|}}}\ Bigg {| |}}}\ Bigg {|} |} _ {n+1} =\ frac {\ mathbb {J}} {3} -\ boldsymbol {N} _ {n+1}\ otimes\ boldsymbol {N} _ {n+1}} {T_ {eq,\ alpha} (\ boldsymbol {N}} _ {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,\ alpha} (\ boldsymbol {A} _ {n+1})} 
  \ end {align}
  :label: derivee_utile_3

First line
--------------

The derivation of each term in the first line of the system shown :eq:`residu_plastique_jacobienne_annexe` -b provides:

.. math::
\ begin {align}  
  &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ partial\ Delta\ boldsymbol {\ epsilon} ^e} =\ mathbb {I} +\ Delta\ lambda_1\ boldsymbol {N} __1} _ {n+1}} {n+1}} =\ mathbb {C} _ {n+1}\\  
  &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ Delta\ lambda_1} =\ boldsymbol {N} _ {n+1}\\
  &\ frac {\ partial\ boldsymbol {r} _1} {\ partial\ delta\ xi_1} =\ Delta\ lambda\ boldsymbol {N} _ {n+1,\ boldsymbol {A} _ {n+1}}}:\ frac {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ delta\ xi_1}}\ Bigg {|} _ {n+1}} -\ cfrac {\ partial\ Delta\ lambda_2} {\ partial\ Delta\ xi_1}\ cfrac {\ boldsymbol {I}}} {3}
  \ end {align}
  :label: derivee_utile_2

with, in accordance with :eq:`ecrouissage_expression` -a and :eq:`expression_delta_lambda2`: 

.. math::
\ frac {\ partial\ boldsymbol {A}} {\ partial\ Delta\ xi_1}\ Bigg {|} _ {n+1} = -\ frac {\ tilde p_ {con} (p_ {con} (p_ {c, n+1}},\ varepsilon_ {v, n} ^p+
  \ Delta\ xi_1-\ Delta\ lambda_2)} {2 (\ tilde\ lambda (p_ {c, n+1}) -\ tilde\ kappa)}\ boldsymbol {I},\ quad 
  \ cfrac {\ partial\ Delta\ delta\ lambda_2} {\ partial\ Delta\ xi_1} = 
  \ begin {cases}
  1\ quad &\ text {si}\ quad f_2 (p_ {c, n+1},\ epsilon_ {v, n} ^p+\ Delta\ xi_1) >0\\
  0\ quad &\ text {otherwise}
  \ end {cases}
  :label: derivees_xi1_1

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

The second line of the system reads:

.. math::
\ begin {align}
 &\ frac {\ partial r_2} {\ partial\ Delta\ delta\ boldsymbol {\ epsilon} ^e} =\ cfrac {\ boldsymbol {M} _ {n+1}:\ mathbb {C} {C} _ {c} _ {n+1}}} {\ mu}\\
 &\ frac {\ partial r_2} {\ partial\ Delta\ lambda_1} =0\\
 &\ frac {\ partial r_2} {\ partial\ boldsymbol {A}} {\ partial\ Delta\ xi_1} =\ cfrac {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ Delta\ xi_1}}\ Bigg {|} _ {n+1}} - 
 \ cfrac {\ partial R} {\ partial\ Delta\ xi_1}\ Bigg {|} _ {n+1}} {\ mu}
 \ end {align}
 :label: jacobienne_line2

where we write:

.. math::
 R (p_c,\ varepsilon_v^p) =\ frac {\ tilde p_ {con} (p_c,\ varepsilon_v^p) +k_sp_c} {2}\ Longrightarrow
 \ frac {\ partial R} {\ partial\ Delta\ xi_1}\ Bigg {|} _ {n+1} = -\ frac {\ tilde p_ {con} (p_ {c, n+1}},\ varepsilon_ {v, n} ^p+
 \ Delta\ xi_1-\ Delta\ lambda_2)} {2 (\ tilde\ lambda (p_ {c, n+1}) -\ tilde\ kappa)}
 :label: derivees_xi1_2


Third line
---------------

Finally, the third line of the system is:

.. math::
\ begin {align}
 &\ frac {\ partial r_3} {\ partial\ Delta\ Delta\ boldsymbol {\ epsilon} ^e} = -\ Delta\ lambda_1 N_ {v, n+1,\ boldsymbol {A} {A} _ {n+1}}}:\ mathbb {C} _ {n+1}\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ lambda_1} =-N_ {v, n+1}\\
 &\ frac {\ partial r_3} {\ partial\ Delta\ xi_1} =1 -\ Delta\ lambda_1 N_ {v, n+1,\ boldsymbol {A} _ {n+1}}}:\ frac {\ partial\ delta\ xi_1}}:\ frac {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}} {\ partial\ boldsymbol {A}}\ Bigg {|} _ {n+1}}:\ frac {\ partial\ boldsymbol {A}}
 \ end {align}
 :label: jacobienne_line3