Iaconeta et al 2020a - Revision history

Cinmemj: Cinmemj moved page Draft Samper 987121664 to Iaconeta et al 2020a

2020-02-13T15:13:22Z

Cinmemj moved page Draft Samper 987121664 to Iaconeta et al 2020a

Cinmemj at 15:08, 13 February 2020

2020-02-13T15:08:52Z

Cinmemj at 14:52, 13 February 2020

2020-02-13T14:52:16Z

Cinmemj at 14:31, 13 February 2020

2020-02-13T14:31:00Z

Cinmemj at 14:26, 13 February 2020

2020-02-13T14:26:51Z

Cinmemj at 13:51, 13 February 2020

2020-02-13T13:51:53Z

Cinmemj at 13:49, 13 February 2020

2020-02-13T13:49:34Z

Cinmemj at 13:41, 13 February 2020

2020-02-13T13:41:00Z

Cinmemj at 13:37, 13 February 2020

2020-02-13T13:37:43Z

Cinmemj at 13:32, 13 February 2020

2020-02-13T13:32:23Z

@@ Line 4,762: / Line 4,762: @@
 ::(b) Evaluation of local Jacobian matrix of residual (<math display="inline">lhs</math>) (LHS of Equation [[#eq-5.17|5.17]])
 ::(c) Assemble rhs and lhs to the global vector <math display="inline">RHS</math> and global matrix <math display="inline">LHS</math> (see the matricial system of Equation [[#eq-5.17|5.17]]))
-:* Solving system <math display="inline"> (^{it+1}\delta \boldsymbol{u}_I^{n+1}, ^{it+1}\delta p_I^{n+1})
+:* Solving system <math display="inline"> (^{it+1}\delta \boldsymbol{u}_I^{n+1}, ^{it+1}\delta p_I^{n+1}) </math>
 :* Newmark method: CORRECTOR by using the Equations <math display="inline">^{it+1}\Delta \boldsymbol{u}_{I}^{n+1} =^{it}\Delta \boldsymbol{u}_{I}^{n+1} +^{it+1}\delta \boldsymbol{u}_I^{n+1}</math>(Equation [[#eq-2.13|2.13]]) and <math display="inline">^{it+1}p_I^{n+1} = ^{it+1}\delta p_I^{n+1}</math>
 :* Check convergence

@@ Line 4,744: / Line 4,744: @@
 |-
+|
 <ol>
 <li>'''INITIALIZATION PHASE''' </li>
 <li>'''INITIALIZATION PHASE'''
-{|
-|-
-| </li>
 :* Clear nodal info and recover undeformed grid configuration
 :* Calculation of initial nodal conditions.
 ::(a) for p = 1:<math display="inline"> N_p </math>
 :::* Calculation of nodal data
 ::::* <math display="inline"> l_I^n = \sum _p N_I \, m_p p_p^n </math>
 ::::* <math display="inline"> m_I^n = \sum _p N_I m_p </math>
 ::(b) for I = 1:<math display="inline"> N_I </math>
 :::* <math display="inline"> \widetilde{p}_{I}^n = \dfrac{l_I^n}{m_I^n}</math>
 :* Newmark method: PREDICTOR. Evaluation of <math display="inline">^{it+1}\Delta \mathbf{u}_I^{n+1} , ^{it+1}p_I^{n+1}</math> by using Equations <math display="inline">^{it+1}\Delta \boldsymbol{u}_{I}^{n+1}= 0.0</math> and <math display="inline">^{it+1}p_I^{n+1}=\widetilde{p}_{I}^n</math>
-<li>'''UL-FEM PHASE'''
+<li>'''UL-FEM PHASE''' </li>
-{|
-|-
-| </li>
 :* for p = 1:<math display="inline"> N_p </math>
 ::(a) Evaluation of local residual (<math display="inline">rhs</math>) (RHS of Equation [[#eq-5.17|5.17]])
 ::(b) Evaluation of local Jacobian matrix of residual (<math display="inline">lhs</math>) (LHS of Equation [[#eq-5.17|5.17]])
 ::(c) Assemble rhs and lhs to the global vector <math display="inline">RHS</math> and global matrix <math display="inline">LHS</math> (see the matricial system of Equation [[#eq-5.17|5.17]]))
-|-
+:* Solving system <math display="inline"> (^{it+1}\delta \boldsymbol{u}_I^{n+1}, ^{it+1}\delta p_I^{n+1})
-−
-|}
-−
-:* Solving system <math display="inline"> (^{it+1}\delta \boldsymbol{u}_I^{n+1}, ^{it+1}\delta p_I^{n+1}) </math>
-|-
-−
 :* Newmark method: CORRECTOR by using the Equations <math display="inline">^{it+1}\Delta \boldsymbol{u}_{I}^{n+1} =^{it}\Delta \boldsymbol{u}_{I}^{n+1} +^{it+1}\delta \boldsymbol{u}_I^{n+1}</math>(Equation [[#eq-2.13|2.13]]) and <math display="inline">^{it+1}p_I^{n+1} = ^{it+1}\delta p_I^{n+1}</math>
 :* Check convergence
 ::(a) NOT converged: go to Step 2
 ::(b) Converged: go to Step 3
 :* Save the stress <math display="inline"> \boldsymbol{\sigma }^{n+1}_p </math>, strain <math display="inline"> \boldsymbol{\epsilon }^{n+1}_p </math>, pressure <math display="inline">p^{n+1}_p</math> and total deformation gradient <math display="inline"> \mathbf{F}^{n+1}_p </math> on material points (the latter by&nbsp;<math display="inline">\mathbf{F}_p^{n+1}=\Delta \mathbf{F}_p \cdot \mathbf{F}_p^n</math>)
 </ol>
 |-
 | style="text-align: center; font-size: 75%;"|

@@ Line 2,480: / Line 2,480: @@
 :* Compute the deviatoric part of the <math display="inline">\boldsymbol{\tau }</math>: <math display="inline">\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}=\mu \mathrm{dev}\left[{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}\right]</math>
 <li>'''CHECK FOR PLASTIC LOADING'''</li>
+:  Evaluate the Mises-Huber yield condition <math display="inline">f^{\mathrm{trial}}_{n+1}:=\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert -\sqrt{\frac{2}{3}}\left(H \alpha _n + \sigma _Y \right)</math>
-;  Evaluate the Mises-Huber yield condition <math display="inline">f^{\mathrm{trial}}_{n+1}:=\lVert \mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}\rVert -\sqrt{\frac{2}{3}}\left(H \alpha _n + \sigma _Y \right)</math>
 {|
 |-
@@ Line 2,501: / Line 2,500: @@
 |}
 <li>'''EVALUATE THE STRESS IN CURRENT CONFIGURATION'''
-; <math display="inline">\boldsymbol{\tau }_{n+1}=J_{n+1}U'(J_{n+1})+ \mathrm{dev}\boldsymbol{\tau }_{n+1} </math> </li>
+: <math display="inline">\boldsymbol{\tau }_{n+1}=J_{n+1}U'(J_{n+1})+ \mathrm{dev}\boldsymbol{\tau }_{n+1} </math> </li>
 <li>'''UPDATE THE INTERMEDIATE CONFIGURATION'''
-; <math display="inline"> {\overline{\boldsymbol{b}}^e}_{n+1}=\dfrac{\mathrm{dev}\boldsymbol{\tau }_{n+1}}{\mu }+\dfrac{1}{3}\mathrm{tr}\left[{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}\right]\boldsymbol{1} </math>  </li>
+: <math display="inline"> {\overline{\boldsymbol{b}}^e}_{n+1}=\dfrac{\mathrm{dev}\boldsymbol{\tau }_{n+1}}{\mu }+\dfrac{1}{3}\mathrm{tr}\left[{\overline{\boldsymbol{b}}^e}^{\mathrm{trial}}_{n+1}\right]\boldsymbol{1} </math>  </li>
 </ol>
 </span>

@@ Line 3,443: / Line 3,443: @@
 </li>
-;  Evaluate the Mohr-Coulomb yield condition <math display="inline">f(\boldsymbol{\tau }^{\mathrm{trial}})=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \phi - 2c \cos \phi </math>
+:  Evaluate the Mohr-Coulomb yield condition <math display="inline">f(\boldsymbol{\tau }^{\mathrm{trial}})=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \phi - 2c \cos \phi </math>
 :a If <math display="inline">f^{\mathrm{trial}}_{n+1}\leq{0}</math>, no plastic loading is observed. Thus, <math display="inline">\boldsymbol{\tau }_{n+1} = \boldsymbol{\tau }^{\mathrm{trial}}</math> and <math display="inline"> {\boldsymbol{b}^e}_{n+1} = {\boldsymbol{b}^e}^{\mathrm{trial}}_{n+1}</math>
@@ Line 3,456: / Line 3,456: @@
 </li>
-;  Transform <math display="inline">\boldsymbol{\tau }_{n+1}</math> and elasto-plastic tangent moduli back to the original coordinate system
+:  Transform <math display="inline">\boldsymbol{\tau }_{n+1}</math> and elasto-plastic tangent moduli back to the original coordinate system
 {|
 <li>'''UPDATE THE INTERMEDIATE CONFIGURATION'''
 </li>
 |}
-;  Update <math display="inline"> {\boldsymbol{b}^e}_{n+1}=\sum \limits _{A=1}^3 exp\left[2 \boldsymbol{\epsilon }^{e}_A \right]\boldsymbol{m}^{A} </math>
+:  Update <math display="inline"> {\boldsymbol{b}^e}_{n+1}=\sum \limits _{A=1}^3 exp\left[2 \boldsymbol{\epsilon }^{e}_A \right]\boldsymbol{m}^{A} </math>
 </ol>
 </span>

@@ Line 3,439: / Line 3,439: @@
 :* Compute the logarithmic principal stretches <math display="inline">\boldsymbol{\epsilon }^{e, \mathrm{trial}} = {{\mathrm{ln}}}(\lambda _A)</math>
 :* Compute the principal trial Kirchhoff stresses <math display="inline">\boldsymbol{\tau }^{\mathrm{trial}}=\mathrm{a}\boldsymbol{\epsilon }^{e, \mathrm{trial}}</math>
-{|
-<li>'''CHECK FOR PLASTIC LOADING'''</li>
+<li>'''CHECK FOR PLASTIC LOADING'''
-|}
+</li>
 ;  Evaluate the Mohr-Coulomb yield condition <math display="inline">f(\boldsymbol{\tau }^{\mathrm{trial}})=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \phi - 2c \cos \phi </math>
@@ Line 3,451: / Line 3,452: @@
 ::* Correct <math display="inline">\boldsymbol{\tau }</math>: <math display="inline">\boldsymbol{\tau }_{n+1}=\boldsymbol{\tau }^{\mathrm{trial}}-\Delta \gamma \mathrm{a}\frac{\delta g(\boldsymbol{\tau }^{\mathrm{trial}})}{\delta \boldsymbol{\tau }}</math>
 ::* Correct <math display="inline">\boldsymbol{\epsilon }^{e}</math>: <math display="inline">\boldsymbol{\epsilon }^{e}=\mathrm{a}^{-1}\boldsymbol{\tau }_{n+1}</math>
-{|
-<li>'''EVALUATE THE STRESS IN CURRENT CONFIGURATION''' </li>
+<li>'''EVALUATE THE STRESS IN CURRENT CONFIGURATION'''
-|}
+</li>
 ;  Transform <math display="inline">\boldsymbol{\tau }_{n+1}</math> and elasto-plastic tangent moduli back to the original coordinate system
 {|
 <li>'''UPDATE THE INTERMEDIATE CONFIGURATION'''
 |}
-;  Update <math display="inline"> {\boldsymbol{b}^e}_{n+1}=\sum \limits _{A=1}^3 exp\left[2 \boldsymbol{\epsilon }^{e}_A \right]\boldsymbol{m}^{A} </math>  </li>
+;  Update <math display="inline"> {\boldsymbol{b}^e}_{n+1}=\sum \limits _{A=1}^3 exp\left[2 \boldsymbol{\epsilon }^{e}_A \right]\boldsymbol{m}^{A} </math>
 </ol>
 </span>

← Older revision	Revision as of 15:13, 13 February 2020
(No difference)

@@ Line 3,428: / Line 3,428: @@
 |-
+|
 <ol>
+<li>'''UPDATE THE CURRENT CONFIGURATION''' </li>
 <li>UPDATE THE CURRENT CONFIGURATION
-{|
-|-
-|   </li>
 :* Compute the the current configuration: <math display="inline"> \boldsymbol{\varphi }_{n+1}=\boldsymbol{\varphi }_n + \boldsymbol{u}_n \circ \boldsymbol{\varphi }_n </math>
 :* Compute the relative deformation gradient: <math display="inline">\boldsymbol{f}_{n+1} = \boldsymbol{1} + \nabla _{\boldsymbol{x}_n} \boldsymbol{u}_n</math>
 :* Compute the total deformation gradient in updated configuration: <math display="inline">\boldsymbol{F}_{n+1} = \boldsymbol{f}_{n+1}\boldsymbol{F}_n</math>
-<li>COMPUTE THE ELASTIC PREDICTOR
+<li>'''COMPUTE THE ELASTIC PREDICTOR'''  </li>
-{|
-|-
-| </li>
 :* Compute the trial elastic left Cauchy-Green tensor of <math display="inline">{\boldsymbol{b}^e}^{\mathrm{trial}}= \boldsymbol{f}_{n+1}{\boldsymbol{b}^e}_n {\boldsymbol{f}_{n+1}}^T</math>
 :* Compute the spectral decomposition of <math display="inline">{\boldsymbol{b}^e}^{\mathrm{trial}}= \sum \limits _{A=1}^3 \lambda _A^2 \boldsymbol{m}^{A}</math>
 :* Compute the logarithmic principal stretches <math display="inline">\boldsymbol{\epsilon }^{e, \mathrm{trial}} = {{\mathrm{ln}}}(\lambda _A)</math>
 :* Compute the principal trial Kirchhoff stresses <math display="inline">\boldsymbol{\tau }^{\mathrm{trial}}=\mathrm{a}\boldsymbol{\epsilon }^{e, \mathrm{trial}}</math>
-<li>CHECK FOR PLASTIC LOADING
+<li>'''CHECK FOR PLASTIC LOADING'''</li>
 ;  Evaluate the Mohr-Coulomb yield condition <math display="inline">f(\boldsymbol{\tau }^{\mathrm{trial}})=(\tau _1 - \tau _3)+(\tau _1 + \tau _3)\sin \phi - 2c \cos \phi </math>
-{|
-|-
-| </li>
 :a If <math display="inline">f^{\mathrm{trial}}_{n+1}\leq{0}</math>, no plastic loading is observed. Thus, <math display="inline">\boldsymbol{\tau }_{n+1} = \boldsymbol{\tau }^{\mathrm{trial}}</math> and <math display="inline"> {\boldsymbol{b}^e}_{n+1} = {\boldsymbol{b}^e}^{\mathrm{trial}}_{n+1}</math>
 :b If <math display="inline">f^{\mathrm{trial}}_{n+1}>0</math>, plastic loading is observed.
-|-
-−
-{|
-|-
-−
 ::* Define the type of return (see Section [[#3.3.1 The stress regions|3.3.1]])
 ::* Applied the corresponding return stress and compute the elasto-plastic tangent moduli (see Section [[#3.3.2 Stress update in principal stress space|3.3.2]])
 ::* Correct <math display="inline">\boldsymbol{\tau }</math>: <math display="inline">\boldsymbol{\tau }_{n+1}=\boldsymbol{\tau }^{\mathrm{trial}}-\Delta \gamma \mathrm{a}\frac{\delta g(\boldsymbol{\tau }^{\mathrm{trial}})}{\delta \boldsymbol{\tau }}</math>
-|-
-−
 ::* Correct <math display="inline">\boldsymbol{\epsilon }^{e}</math>: <math display="inline">\boldsymbol{\epsilon }^{e}=\mathrm{a}^{-1}\boldsymbol{\tau }_{n+1}</math>
-<li>EVALUATE THE STRESS IN CURRENT CONFIGURATION
+<li>'''EVALUATE THE STRESS IN CURRENT CONFIGURATION''' </li>
-;  Transform <math display="inline">\boldsymbol{\tau }_{n+1}</math> and elasto-plastic tangent moduli back to the original coordinate system </li>
+;  Transform <math display="inline">\boldsymbol{\tau }_{n+1}</math> and elasto-plastic tangent moduli back to the original coordinate system
 <li>UPDATE THE INTERMEDIATE CONFIGURATION
 ;  Update <math display="inline"> {\boldsymbol{b}^e}_{n+1}=\sum \limits _{A=1}^3 exp\left[2 \boldsymbol{\epsilon }^{e}_A \right]\boldsymbol{m}^{A} </math>  </li>
 </ol>
 </span>
 |-
 | style="text-align: center; font-size: 75%;"|

@@ Line 2,496: / Line 2,496: @@
 |} Computation of the return mapping
 |-
-| Correct
+| Correct <math display="inline">\mathrm{dev}\boldsymbol{\tau }</math>: <math display="inline">\mathrm{dev}\boldsymbol{\tau }_{n+1}=\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}-2\overline{\mu }\Delta \gamma \boldsymbol{n}</math>
-−
 <math display="inline">\mathrm{dev}\boldsymbol{\tau }</math>: <math display="inline">\mathrm{dev}\boldsymbol{\tau }_{n+1}=\mathrm{dev}\boldsymbol{\tau }^{\mathrm{trial}}-2\overline{\mu }\Delta \gamma \boldsymbol{n}</math>
 |-
 | Correct <math display="inline">\alpha _{n+1}</math>: <math display="inline">\alpha _{n+1} = \alpha _n+\sqrt{\frac{2}{3}}\Delta \gamma  </math>