8. Appendix: Expression of the Jacobian matrix#
In the case where the plasticity criterion \(f_1\) is not 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\ r_2\ r_3end {Bmatrix} =
begin {Bmatrix} Deltaboldsymbol {varepsilon} ^e -Deltaboldsymbol {varepsilon} +Deltalambda_1cfrac {partial g_1} {partialboldsymbol {sigma} « }Bigg {sigma} »}Bigg {|} _ {n+1} -Deltalambda_1cfrac {partial g_1}} {partialboldsymbol {sigma} « } »}Bigg {|} _ {n+1} -Deltalambda_2cfrac {boldsymbol {I}} {3}\ cfrac {f_ {1, n+1}} {mu}\ Deltaxi_1-Deltalambda_1boldsymbol {I}:cfrac {partial g_1} {partialboldsymbol {sigma} « }Bigg {|}Bigg {|} |} _ {n+1}\ end {Bmatrix},quad J =begin {Bmatrix} cfrac {partialboldsymbol {r} _1} {partialDeltaboldsymbol {varepsilon} ^e}} & cfrac {partialboldsymbol {r} _1} {partialDeltalambda_1}} & cfrac {partialboldsymbol {r} _1} {partialDeltaxi_1}\ cfrac {partial r_2} {partialDeltaDeltaboldsymbol {varepsilon} ^e}} & cfrac {partial r_2} {partialDeltalambda_1}} & cfrac {partial r_2} {partialDeltaxi_1}\ cfrac {partial r_3} {partialDeltaDeltaboldsymbol {varepsilon} ^e}} & cfrac {partial r_3} {partialDeltalambda_1}} & cfrac {partial r_3} {partialDeltaxi_1} end {Bmatrix} :label: Jacobienne_plastic_residure_appendix
In MFront, the \(J\) 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:
- boldsymbol {M} _ {n+1} =frac {partial f_1} {partialboldsymbol {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})})},
quadtext {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 \(\Delta\boldsymbol{\varepsilon}_1^p\) and \(\Delta\xi_1\):
- begin {align}
&boldsymbol {N} _ {n+1} =frac {cfrac {partial g_1} {partialboldsymbol {sigma} « }Bigg {|} _ {n+1} =frac {cfrac {cfrac {3frac {3alpha} {3alpha} {2M^2} {2M^2}boldsymbol {A} _ {d, n+1} +A_ {m, n+1} =frac {cfrac {cfrac {3alpha} {2M^2}cfrac {boldsymbol {I}} {3}} {3}} {T_ {eq,alpha} (boldsymbol {A} _ {n+1})})}, quadtext {with}quad T_ {eq,alpha} (boldsymbol {A}) =sqrt {alphaleft (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:
- begin {align}
&boldsymbol {N} _ {n+1,boldsymbol {A} _ {n+1}} =frac {partialboldsymbol {N}} {partialboldsymbol {A}}}Bigg {|}}}Bigg {|} _ {n+1} =frac {cfrac {3partialboldsymbol {N}}} {partialboldsymbol {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}otimesboldsymbol {N} _ {n+1}} {T_ {eq,alpha} (boldsymbol {N}} _ {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,alpha} (boldsymbol {A} _ {n+1})} end {align} :label: derivee_utile_3
8.1. First line#
The derivation of each term in the first line of the system shown residu_plastique_jacobienne_annexe -b provides:
- begin {align}
&frac {partialboldsymbol {r} _1} {partialpartialDeltaboldsymbol {epsilon} ^e} =mathbb {I} +Deltalambda_1boldsymbol {N} __1} _ {n+1}} {n+1}} =mathbb {C} _ {n+1}\ &frac {partialboldsymbol {r} _1} {partialDeltalambda_1} =boldsymbol {N} _ {n+1}\ &frac {partialboldsymbol {r} _1} {partialdeltaxi_1} =Deltalambdaboldsymbol {N} _ {n+1,boldsymbol {A} _ {n+1}}}:frac {partialboldsymbol {A}} {partialboldsymbol {A}} {partialdeltaxi_1}}Bigg {|} _ {n+1}} -cfrac {partialDeltalambda_2} {partialDeltaxi_1}cfrac {boldsymbol {I}}} {3} end {align} :label: derivee_utile_2
with, in accordance with ecrouissage_expression -a and expression_delta_lambda2:
- frac {partialboldsymbol {A}} {partialDeltaxi_1}Bigg {|} _ {n+1} = -frac {tilde p_ {con} (p_ {con} (p_ {c, n+1}},varepsilon_ {v, n} ^p+
Deltaxi_1-Deltalambda_2)} {2 (tildelambda (p_ {c, n+1}) -tildekappa)}boldsymbol {I},quad cfrac {partialDeltadeltalambda_2} {partialDeltaxi_1} = begin {cases} 1quad &text {si}quad f_2 (p_ {c, n+1},epsilon_ {v, n} ^p+Deltaxi_1) >0\ 0quad &text {otherwise} end {cases} :label: derivees_xi1_1
8.2. Second line#
The second line of the system reads:
- begin {align}
&frac {partial r_2} {partialDeltadeltaboldsymbol {epsilon} ^e} =cfrac {boldsymbol {M} _ {n+1}:mathbb {C} {C} _ {c} _ {n+1}}} {mu}\ &frac {partial r_2} {partialDeltalambda_1} =0\ &frac {partial r_2} {partialboldsymbol {A}} {partialDeltaxi_1} =cfrac {partialboldsymbol {A}} {partialboldsymbol {A}} {partialDeltaxi_1}}Bigg {|} _ {n+1}} - cfrac {partial R} {partialDeltaxi_1}Bigg {|} _ {n+1}} {mu} end {align} :label: jacobienne_line2
where we write:
8.3. Third line#
Finally, the third line of the system is:
- begin {align}
&frac {partial r_3} {partialDeltaDeltaboldsymbol {epsilon} ^e} = -Deltalambda_1 N_ {v, n+1,boldsymbol {A} {A} _ {n+1}}}:mathbb {C} _ {n+1}\ &frac {partial r_3} {partialDeltalambda_1} =-N_ {v, n+1}\ &frac {partial r_3} {partialDeltaxi_1} =1 -Deltalambda_1 N_ {v, n+1,boldsymbol {A} _ {n+1}}}:frac {partialdeltaxi_1}}:frac {partialboldsymbol {A}} {partialboldsymbol {A}} {partialboldsymbol {A}} {partialboldsymbol {A}} {partialboldsymbol {A}} {partialboldsymbol {A}} {partialboldsymbol {A}}Bigg {|} _ {n+1}}:frac {partialboldsymbol {A}} end {align} :label: jacobienne_line3