6. Appendix: Expression of the Jacobian matrix#
In the case where neither the plasticity criterion \(f\) nor \(F_m\) is verified after the elastic prediction, we recall the expression of the system
of nonlinear equations residu_plastique and the definition of the Jacobian matrix jacobienne:
- begin {Bmatrix}boldsymbol {r} _1\boldsymbol {r} _2\ r_3\ r_4\ r_5end {Bmatrix} =
begin {Bmatrix} Deltaboldsymbol {varepsilon} ^e -Deltaboldsymbol {varepsilon} + left (mathbb {J} +rhomathbb {K}right):Deltalambdacfrac {partial f} {partialboldsymbol {X}}big {|} _ {n+1}} + left (mathbb {J} + (1-rho)mathbb {K}right):left (Deltalambda_mcfrac {partial F_m} {partial F_m} {partialboldsymbol {A} ^m}big {|} _ {n+1}} +sum_ {i=1} ^ {m-1}Deltaboldsymbol {alpha} ^iright)\ Deltaboldsymbol {varepsilon} ^x -Deltaboldsymbol {varepsilon} ^e + left (1-rhoright)mathbb {K}:left (DeltaLambdacfrac {partial f} {partialboldsymbol {X}}}Big {|} {K}}Big {|} _ {n+1} -Deltalambda_mcfrac {partial F_m} {partialboldsymbol {A} ^m}Big {|} _ {n+1} -sum_ {i=1} ^ {m-1}Deltaboldsymbol {alpha} ^iright)\ cfrac {f_ {n+1}} {K}\ cfrac {F_ {m, n+1}} {K}\ Deltaxi -DeltaLambdacfrac {partial f} {partial p_c}Big {|} _ {n+1}\ end {Bmatrix} :label: plastic_residure_appendix
In MFront, this \(J\) matrix can be obtained by numerical disturbance or analytically, as is the case presented below. Its components are detailed below.
For this purpose, note:
The flow directions of \(\Delta\boldsymbol{\varepsilon}^p,\left(\Delta\boldsymbol{\alpha}^i\right)_{1\leq i\leq m}\) and \(\Delta\xi\):
- begin {align}
&boldsymbol {N} _ {n+1} =frac {partial f} {partialboldsymbol {X}}big {|} _ {n+1} =frac {cfrac {3} {3} {2M^2} =frac {3} {2M^2} {2M^2}}boldsymbol {2}}boldsymbol {2}}boldsymbol {Y} _ {d, n+1} +Y_ {m, n+1} =frac {cfrac {3} {2M^2} {2M^2}}boldsymbol {2}}frac {boldsymbol {I}} {3}} {T_ {eq} (boldsymbol {Y} _ {n+1})} ,quadtext {with}quadboldsymbol {Y} =boldsymbol {X} +left (P_c-sright)boldsymbol {I},quad T_ {eq} (boldsymbol {Y}) (boldsymbol {Y}) =sqrt {Y}) =sqrt {Y}) =sqrt {X} +left (frac {Y_ {eq}} {M}right) ^2+y_M^2}right) ^2+y_M^2}\ &boldsymbol {M} ^i_ {n+1} =frac {partial F_i} {partialboldsymbol {A} ^i}Big {|} _ {n+1} =frac {cfrac {3} {2} =frac {cfrac {3} {2} {2}boldsymbol {A} ^i_ {d, n+1}} +left (cfrac {R_i}} {C}right) ^2leftlangle A^i_ {m, n+1} +Crightranglecfrac {boldsymbol {I}} {3}} {t^i_ {eq}} {t^i_ {eq} (boldsymbol {A} ^i_ {n+1})} ,quadtext {with}quad T^i_ {eq} (boldsymbol {A} ^i) =sqrt {left (A^i_ {eq}right) ^2+left (cfrac {R_i} {eq} {eq} {eq}} (boldsymbol {A} ^i) =sqrt {left (A^i_ {eq}right) ^2+left (cfrac {R_i} {eq} {eq}left (cfrac {R_i}} {C}leftlangle A^i_ {m} +Crightrangleright) ^2}\ right &N_ {v, n+1} =frac {partial f} {partial p_c}Big {|} _ {n+1} =boldsymbol {I}:boldsymbol {I}:boldsymbol {N} _ {n+1}\ end {align} :label: derivee_utile_1
Their following three derivatives:
- begin {align}
&boldsymbol {N} _ {n+1,boldsymbol {Y} _ {n+1}} =frac {partialboldsymbol {N}} {partialboldsymbol {Y}}Big {|}big {|} _ {n+1} =frac {n+1} =frac {cfrac {3} {2M^2}}mathbb {K} +cfrac {mathbb} +cfrac {mathbb {J}} {3} -boldsymbol {N} _ {n+1} _ {n+1}otimesboldsymbol {N} _ {n+1}} {T_ {eq}} (boldsymbol {Y} _ {n+1})}\ &boldsymbol {M} ^i_ {n+1,boldsymbol {A} ^i_ {n+1}} =frac {partialboldsymbol {M} ^i} {partialboldsymbol {A} ^i}big {|} _ {n+1}} = frac {cfrac {3} {2}mathbb {K} +left (cfrac {R_i} {C}right) ^2mathbf {H}left (A^i_ {m, n+1} {m, n+1} +Cright} +Cright)cfrac {mathbb {J}} {3} -boldsymbol {M} ^i_ {n+1}otimesboldsymbol {M} ^i_ {n+1}} {t^i_ {eq} (boldsymbol {A} ^i_ {n+1})}\ &N_ {v, n+1,boldsymbol {Y} _ {n+1}} =frac {partial N_ {v}} {partialboldsymbol {Y}}big {|} _ {n+1} =frac {cfrac {cfrac {boldsymbol {I}}} {3} -N_ {v, n+1}}big {|} _ {n+1} =frac {cfrac {boldsymbol {I}}} {3} -N_ {v, n+1}}big {|} _ {n+1} =frac {cfrac {boldsymbol {I}}} {3} -N_ {v, n+1}}big {|}} _ {n+1} =frac {n+1}} {T_ {eq} (boldsymbol {Y} _ {n+1})} end {align} :label: derivee_utile_2
Derivatives of forces \(\boldsymbol{Y}=\boldsymbol{Y}(\boldsymbol{\varepsilon}^x,\lambda,\xi)\) and \(\left(\boldsymbol{A}^i\right)_{1\leq i \leq m}=\left(\boldsymbol{A}^i(\boldsymbol{\varepsilon}^e,\boldsymbol{\varepsilon}^x)\right)_{1\leq i \leq m}\):
- begin {align}
&boldsymbol {Y} _ {n+1,boldsymbol {varepsilon} ^x} =frac {partialboldsymbol {Y}} {partialDeltaboldsymbol {boldsymbol {varepsilon} ^x} ^x}big {|} ^x}big {|} _ {n+1} =left (mathbb {J}} +rhomathbb {K}right):mathbb {C}\ &boldsymbol {Y} _ {n+1,lambda} =frac {partialboldsymbol {Y}} {partialDeltalambda}Big {|} _ {n+1} =omega S_ {n+1} =omega S_ {n+1}\ &boldsymbol {Y} _ {n+1,xi} =frac {partialboldsymbol {Y}} {partialDeltaxi}Big {|} _ {n+1} = -left (left (omega S_ {n+1}} +beta p_ {n+1}}}right)boldsymbol {I}\ &boldsymbol {A} ^i_ {n+1,boldsymbol {varepsilon} ^e} =frac {partialboldsymbol {A} ^i} {partialDeltaDeltadeltaboldsymbol {boldsymbol {varepsilon} ^e}\&boldsymbol {A} ^i_ {n+1,boldsymbol {varepsilon} ^x} =frac {partialboldsymbol {A} ^i} {partialDeltaDeltaboldsymbol {varepsilon} boldsymbol {boldsymbol {varepsilon} ^x}}big {|} |} _ {n+1} = -rhomathbb {K}:mathbb {K}:mathbb {C} end {align} :label: derivee_utile_3
Derivatives of all \(\left(\Delta\boldsymbol{\alpha}^i=\Delta\lambda_i\boldsymbol{M}_{n+1}^i\right)_{1\leq i\leq m-1}\):
- begin {align}
frac {partialDeltaboldsymbol {alpha} ^i} {partialDeltaboldsymbol {varepsilon} ^e} =chi_ileft (Deltalambda_iboldsymbol {alpha} {alpha} ^i_ {i} ^i} ^i} ^i_ {n+1}} =chi_ileft (Deltalambda_iiiboldsymbol {alpha} ^i} ^i_ {n+1}} =frac {2} {3H_d_}} +frac {2} {3H_d_d+ ^i}boldsymbol {M} ^i_ {n+1}otimesboldsymbol {M} ^i_ {n+1}right):boldsymbol {A} ^i_ {n+1,boldsymbol {varepsilon} ^e}\ frac {partialDeltaboldsymbol {alpha} ^i} {partialDeltaboldsymbol {varepsilon} ^x} =chi_ileft (Deltalambda_iboldsymbol {alpha} {alpha} ^i_ {i} ^i} ^i_ {n+1}}} =chi_ileft (DeltaLambda_iiiiiiiiiiiiiiiboldsymbol {alpha} ^i} ^i} ^i_ {n+1}} =frac {2} {3H_d_d+1}} +frac {2} {3H_d_d+ ^i}boldsymbol {M} ^i_ {n+1}otimesboldsymbol {M} ^i_ {n+1}right):boldsymbol {A} ^i_ {n+1,boldsymbol {varepsilon} ^x} end {align} :label: derivee_utile_4
where \(\chi_i=1\) if \(\left \langle F_i\left(\boldsymbol{\sigma}_{n+1}-\boldsymbol{X}_{d,n+1} -H_d^i\boldsymbol{\alpha}_{d,n}^i-H_v^i\alpha_{v,n}^i\boldsymbol{I}\right)\right \rangle>0\), zero otherwise.
6.1. 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 {varepsilon} ^e} = mathbb {I} +left (mathbb {J} + (1-rho)mathbb {K}right):left (Deltalambda_mboldsymbol {M} ^m_ {n+1,boldsymbol {N+1} + (1-rho) + (1-rho)mathbb {K}right):left (DeltaLambda_mboldsymbol {M} ^m_ {n+1,m_ {n+1,),boldsymbol {v+1, arepsilon} ^e} +sum_ {i=1} ^ {m-1}frac {partialDeltaboldsymbol {alpha} ^i} {partialDeltaboldsymbol {varepsilon} ^e}right)\ &frac {partialboldsymbol {r} _1} {partialDeltaboldsymbol {varepsilon} ^x} = left (mathbb {J} +rhomathbb {K}right):Deltalambdaboldsymbol {N} _ {n+1,boldsymbol {Y} _ {n+1}}}:boldsymbol {n+1}}}:boldsymbol {N} _ {n+1,boldsymbol {varepsilon} ^x}} +left (mathbb {J} + (1-rho)mathbb {K}right):left (Deltalambda_mboldsymbol {M} ^m_ {n+1,boldsymbol {A} {n+1),boldsymbol {A} {n+1}}:boldsymbol {A} ^m_ {n+1,boldsymbol {varepsilon} ^n+1,boldsymbol {varepsilon} ^m+ x} +sum_ {i=1} ^ {m-1}frac {partialDeltaboldsymbol {alpha} ^i} {partialDeltaboldsymbol {varepsilon} {varepsilon} ^x}right)\ &frac {partialboldsymbol {r} _1} {partialDeltalambda} =left (mathbb {J} +rhomathbb {K}right): left (boldsymbol {N} _ {n+1}} +Deltalambdaboldsymbol {N} _ {n+1,boldsymbol {Y} _ {n+1}}}:boldsymbol {Y}}} +deltalambda}right)\ &frac {partialboldsymbol {r} _1} {partialDeltalambda_m} =left (mathbb {J} + (1-rho)mathbb {K}right):boldsymbol {M} ^m_ {n+1}\ &frac {partialboldsymbol {r} _1} {partialDeltaxi} =left (mathbb {J} +rhomathbb {K}right): Deltalambdaboldsymbol {N} _ {n+1,boldsymbol {Y} _ {n+1}}:boldsymbol {Y} _ {n+1,xi}}:boldsymbol {Y} _ {n+1,xi} end {align} :label: jacobienne_line1
6.2. Second line#
The second line of the system reads:
- begin {align}
&frac {partial r_2} {partialDeltaDeltaboldsymbol {varepsilon} ^e} = -mathbb {I} - (1-rho)mathbb {K}: left (Deltalambda_mboldsymbol {M} ^m_ {n+1,boldsymbol {A} ^m_ {n+1}}:boldsymbol {A} ^m_ {n+1,boldsymbol {M} {n+1,boldsymbol {varepsilon} ^e}} +sum_ {i=1} ^ {m-1}frac {partialsymbol {n+1}frac {partialsymbol {n+1}frac {partialsymbol {n+1}frac {partialsymbol {n+1] +1,boldsymbol {partialDeltaboldsilon} ^e} +sum_ {i=1} ^ {m-1}}frac {partialsymbol {n+1}{alpha} ^i} {partialDeltaDeltaboldsymbol {varepsilon} ^e}right)\ &frac {partial r_2} {partialDeltaDeltaboldsymbol {varepsilon} ^x} =mathbb {I} + (1-rho)mathbb {K}:left ( Deltalambdaboldsymbol {N} _ {n+1,boldsymbol {Y} _ {n+1}}:boldsymbol {Y} _ {n+1,boldsymbol {varepsilon} ^x}} - Deltalambda_mboldsymbol {M} ^m_ {n+1,boldsymbol {A} ^m_ {n+1}}:boldsymbol {A} ^m_ {n+1,boldsymbol {varepsilon} ^m_ {m_} {n+1}}:boldsymbol {partialn+1}frac {partialDeltaboldsymbol {alpha}} ^i} {partialDeltaDeltaboldsymbol {varepsilon} ^x}right)\ &frac {partial r_2} {partialDeltalambda} {partialdeltalambda} = (1-rho)mathbb {K}:left (boldsymbol {N} _ {n+1}} +Deltalambdalambda\ boldsymbol\ lambda\ boldsymbol {N}} +deltalambda\lambdaboldsymbol {N}Lambda}right)\ &frac {partial r_2} {partialDeltadeltalambda_m} =- (1-rho)mathbb {K}:boldsymbol {M} ^m_ {n+1}&frac {partial r_2} {partialDeltaxi}} = (1-rho)mathbb {K}:Deltalambdaboldsymbol {N} _ {n+1,n+1,boldsymbol {Y}} _ {n+1}}:boldsymbol {Y} _ {n+1,xi} end {align} :label: jacobienne_line2
6.3. Third line#
The third line in the system is:
- begin {align}
&frac {partial r_3} {partialDeltaDeltaboldsymbol {varepsilon} ^e} =boldsymbol {0}\ &frac {partial r_3} {partialpartialDeltadeltaboldsymbol {varepsilon} ^x} =cfrac {boldsymbol {N} _ {n+1}:boldsymbol {Y} _ {n+1}\ &frac {partial r_3} {partialDeltalambda} =cfrac {boldsymbol {N} _ {n+1}:boldsymbol {Y} _ {n+1,n+1,lambda} -lambda} -omega S_ {n+1}} {K}\ &frac {partial r_3} {partialDeltalambda_m} =0\ &frac {partial r_3} {partialDeltaxi} =cfrac {boldsymbol {N} _ {n+1}:boldsymbol {Y} _ {n+1,xi} +omega S_ {n+1} +omega S_ {n+1} +beta R_ {n+1} +beta R_ {n+1}\end {align} :label: jacobienne_line3
6.4. Fourth line#
The fourth line:
- begin {align}
&frac {partial r_4} {partialDeltaDeltaboldsymbol {varepsilon} ^e} =cfrac {boldsymbol {M} ^m_ {n+1}:boldsymbol {A} {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1} &frac {partial r_4} {partialDeltaDeltaboldsymbol {varepsilon} ^x} =cfrac {boldsymbol {M} ^m_ {n+1}:boldsymbol {A} {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1}:boldsymbol {A} ^m_ {n+1} &frac {partial r_4} {partialDeltalambda} =0\ &frac {partial r_4} {partialDeltalambda_m} =0\ &frac {partial r_4} {partialDeltaxi} =0\ end {align} :label: jacobienne_line4
6.5. Fifth line#
The fifth and final line:
- begin {align}
&frac {partial r_5} {partialDeltaDeltaboldsymbol {varepsilon} ^e} =boldsymbol {0}\ &frac {partial r_5} {partialDeltaDeltaboldsymbol {varepsilon} ^x} = -Deltalambda N_ {v, n+1,boldsymbol {Y} _ {n+1} _ {n+1}}boldsymbol {Y} _ {n+1}}\ &frac {partial r_5} {partialDeltalambda} =-N_ {v, n+1} -Deltalambda N_ {v, n+1,boldsymbol {Y} _ {n+1}}boldsymbol {Y} _ {n+1}}boldsymbol {Y} _ {n+1,lambda}\ &frac {partial r_5} {partialDeltalambda_m} =0\ &frac {partial r_5} {partialDeltaxi} =1-Deltalambda N_ {v, n+1,boldsymbol {Y} _ {n+1}}boldsymbol {n+1}}boldsymbol {Y} _ {n+1,xi}\ end {align} :label: jacobienne_line5