Comellas et al 2016a - Revision history

Scipediacontent at 12:33, 14 December 2018

2018-12-14T12:33:41Z

Scipediacontent at 12:31, 14 December 2018

2018-12-14T12:31:12Z

Scipediacontent at 09:52, 11 December 2018

2018-12-11T09:52:38Z

Cinmemj: Cinmemj moved page Draft Samper 593843246 to Comellas et al 2016a

2018-03-15T13:52:12Z

Cinmemj moved page Draft Samper 593843246 to Comellas et al 2016a

Cinmemj at 13:44, 15 March 2018

2018-03-15T13:44:46Z

Cinmemj at 13:40, 15 March 2018

2018-03-15T13:40:00Z

Cinmemj at 13:32, 15 March 2018

2018-03-15T13:32:10Z

Cinmemj at 13:24, 15 March 2018

2018-03-15T13:24:50Z

Cinmemj at 13:12, 15 March 2018

2018-03-15T13:12:02Z

Cinmemj at 13:06, 15 March 2018

2018-03-15T13:06:52Z

@@ Line 3,479: / Line 3,479: @@
 <li>Calculate <math display="inline">\theta </math> (per unit of volume), <math display="inline">\theta =N_{i}-\vartheta ^{ant \, 3}</math>.      </li>
 <li>Convert <math display="inline">\theta </math> to the increment of stretch allowed per unit of length,
-<math display="inline">{\Delta \theta _{l} = \sqrt=3 -1}</math>.                         </li>
+<math display="inline">{\Delta \theta _{l} = \sqrt[3]{\theta{+1}} -1}</math>.                         </li>
 <li><code>If <math>\left(\Delta \theta _{l} < \left|\vartheta - \vartheta ^{ant}\right|\right)</math> then</code> <math display="inline">\Delta \vartheta = \Delta \theta _{l}</math>.        </li>

@@ Line 3,159: / Line 3,159: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbf{F}^{g}=\sqrt=3 \, \mathbf{I},    </math>
+| style="text-align: center;" | <math>\mathbf{F}^{g}=\sqrt[3]{\vartheta } \, \mathbf{I},  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.11)

@@ Line 2,409: / Line 2,409: @@
 |}
-where the Lambert <math display="inline">\mathcal=x</math> function is defined as <math display="inline">x=\mathcal{W} \,\exp \left[\mathcal{W}\right]</math>.
+where the Lambert <math display="inline">\mathcal{W}[x]</math> function is defined as <math display="inline">x=\mathcal{W} \,\exp \left[\mathcal{W}\right]</math>.
 So, in the numerical integration scheme of Table&nbsp;[[#table-5|5]]  the damage variable will no longer be initialized to zero  but to the value <math display="inline">D_{0}</math>. And the initial value of the maximum reached value of the damage threshold <math display="inline">\tau ^{max}</math>  will be calculated according to the equations above. The initial damage option has been implemented in PLCd&nbsp;<span id='citeF-1'></span>[[#cite-1|[1]]] in such   a way that different elements can be assigned different values of initial damage ranging from 0 to 1 (both included). However, in the  sake of simplicity, all integration points in a given element are assigned the exact same value.

@@ Line 3,962: / Line 3,962: @@
 <li>Compute the undamaged deviatoric part of the specific strain energy density function <math display="inline">\widetilde{\Psi }_{0}</math>       [ [[#eq-2.60|2.60]] for the Ogden partic.] and the present damage threshold <math display="inline">\tau </math> according       to the Simo and Ju criterion&nbsp;[[#eq-2.84|2.84]].      </li>
-<code>If (<math>\tau > \tau _{max}</math>) then</code> (''damage progresses'')
+;<code>If (<math>\tau > \tau _{max}</math>) then</code> (''damage progresses'')
 :Compute the mechanical damage increment, <math>\Delta D=\left\langle D^{n+1}-D^{n}\right\rangle</math>,             with <math>D^{n+1}</math> from [[#eq-3.42|3.42]]. Obtain <math>\partial D/\partial \tau </math> from [[#eq-3.49|3.49]].
 :<code>If (<math>D^{n+1}<D_{aux}^{n}</math>)  then</code>  (''elastic unloading'')
@@ Line 3,968: / Line 3,968: @@
 :<code>end</code>
 :Update the auxiliary damage variable from the previous step, <math>{D_{aux}^{n}=D^{n+1}}</math>.
-*<code>else   (no further damage)</code>
+;<code>else   (no further damage)</code>
 :Assign <math>\Delta D=0</math> and <math>\partial D/\partial \tau=0</math>.
-*<code>end</code>
+;<code>end</code>
 <li>Evaluate the effective damage and the derivative of the healing variable, <math display="inline">D_{eff}^{n+1}=\left(D_{eff}^{n}+\Delta D+k\,\xi \,\Delta t\right)/\left(1+k\,\Delta t\right)</math>     and <math display="inline">\partial R/\partial D_{eff}=k\,\Delta t</math>.      </li>
-<code>If (<math>(D_{eff}^{n+1}-\xi )<0</math>)  then  (''no further healing'')</code>
+;<code>If (<math>(D_{eff}^{n+1}-\xi )<0</math>)  then  (''no further healing'')</code>
 :Assign <math>D_{eff}^{n+1}=D_{eff}^{n}+\Delta D</math> and <math>\partial R/\partial D_{eff}=0</math>.
-<code>end</code>
+;<code>end</code>
 (continue in the next table)

@@ Line 3,952: / Line 3,952: @@
 :Effective damage, <math>D_{eff}^{n+1}=D_{eff}^{n}=0</math> and mechanical damage, <math>D^{n}=D_{aux}^{n}=0</math>.
 :Maximum reached value of the damage threshold, <math>\tau ^{max}=\tau ^{d}_{0}</math>.
 Algorithm at each load increment <math>n</math>
 :Given: deformation gradient tensor <math>\mathbf{F}</math>, elemental pressure <math>p</math>,       hyperelastic material properties, damage material properties <math>\tau ^{d}_{0}</math> and <math>g_{f}^{d}</math>,      and healing parameters <math>k</math> and&nbsp;<math>\xi </math>.
 <ol>
 <li>Compute the right Cauchy-Green deformation tensor <math display="inline">\mathbf{C}=\mathbf{F}^{T} \cdot \mathbf{F}</math>       and its inverse <math display="inline">\mathbf{C}^{-1}</math>.        </li>
 <li>Calculate the volumetric and deviatoric parts of the predictor hyperelastic stress <math display="inline">\mathbf{S}_{h}=\mathbf{S}_{vol} + \widetilde{\mathbf{S}_{0}}</math>, from the constitutive equation        [ [[#eq-2.103|2.103]] for the Ogden partic.].        </li>
@@ Line 3,967: / Line 3,963: @@
 <code>If (<math>\tau > \tau _{max}</math>) then</code> (''damage progresses'')
 :Compute the mechanical damage increment, <math>\Delta D=\left\langle D^{n+1}-D^{n}\right\rangle</math>,             with <math>D^{n+1}</math> from [[#eq-3.42|3.42]]. Obtain <math>\partial D/\partial \tau </math> from [[#eq-3.49|3.49]].
 :<code>If (<math>D^{n+1}<D_{aux}^{n}</math>)  then</code>  (''elastic unloading'')
@@ Line 3,973: / Line 3,968: @@
 :<code>end</code>
 :Update the auxiliary damage variable from the previous step, <math>{D_{aux}^{n}=D^{n+1}}</math>.
-<code>else   (no further damage)</code>
+*<code>else   (no further damage)</code>
 :Assign <math>\Delta D=0</math> and <math>\partial D/\partial \tau=0</math>.
-<code>end</code>
+*<code>end</code>
 <li>Evaluate the effective damage and the derivative of the healing variable, <math display="inline">D_{eff}^{n+1}=\left(D_{eff}^{n}+\Delta D+k\,\xi \,\Delta t\right)/\left(1+k\,\Delta t\right)</math>     and <math display="inline">\partial R/\partial D_{eff}=k\,\Delta t</math>.      </li>
 <code>If (<math>(D_{eff}^{n+1}-\xi )<0</math>)  then  (''no further healing'')</code>
 :Assign <math>D_{eff}^{n+1}=D_{eff}^{n}+\Delta D</math> and <math>\partial R/\partial D_{eff}=0</math>.
 <code>end</code>
-(continued in the next table)
+(continue in the next table)
 </ol>
 |}

← Older revision	Revision as of 13:52, 15 March 2018
(No difference)

@@ Line 3,451: / Line 3,451: @@
 :Calculate the maximum growth and atrophy rates, <math display="inline">\dot{\vartheta }_{max}^{+}= T_{max}^{+} \vartheta / 3</math> and <math display="inline">\dot{\vartheta }_{max}^{-}= T_{max}^{-} \vartheta / 3</math>.
 :<code>If</code> (<math>Tr\left(\boldsymbol{\sigma }\right)>\sigma _{eq}^{*+}</math>) <code>then</code> (''growth'')
-:<math>\left\{ \begin{array}{c} {l l}                                  g={k}^{+}\left(Tr\left(\boldsymbol{\sigma }\right)- \sigma _{eq}^{*+} \right) &                                                \textrm{ if } {k}^{+}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*+} \right) \le                                                  \dot{\vartheta }_{max}^{+}\\                                 g=\dot{\vartheta }_{max}^{+}                                                                         &                                                \textrm{ if } {k}^{+}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*+} \right) >                                                \dot{\vartheta }_{max}^{+}                                  \end{array} \right.</math>
+:<math>\left\{ \begin{array}{ll}                                  g={k}^{+}\left(Tr\left(\boldsymbol{\sigma }\right)- \sigma _{eq}^{*+} \right) &                                                \hbox{ if } {k}^{+}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*+} \right) \le                                                  \dot{\vartheta }_{max}^{+}\\                                 g=\dot{\vartheta }_{max}^{+}                                                                         &                                                \hbox{ if } {k}^{+}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*+} \right) >                                                \dot{\vartheta }_{max}^{+}                                  \end{array} \right.</math>
-:<code>else if (<math>Tr\left(\boldsymbol{\sigma }\right)<\sigma _{eq}^{*-}</math>) then</code>(''atrophy'')      <math>\left\{ \begin{array}{c} {l l}                                  g={k}^{-}\left(Tr\left(\boldsymbol{\sigma }\right)- \sigma _{eq}^{*-} \right) &                                                        \textrm{ if } {k}^{-}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*-} \right) \ge                                                        \dot{\vartheta }_{max}^{-}\\                                 g=\dot{\vartheta }_{max}^{-}                                                                         &                                                          \textrm{ if } {k}^{-}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*-} \right) <                                                          \dot{\vartheta }_{max}^{-}                                  \end{array} \right.</math>
+:<code>else if (<math>Tr\left(\boldsymbol{\sigma }\right)<\sigma _{eq}^{*-}</math>) then</code>(''atrophy'')      <math>\left\{ \begin{array}{ll}                                  g={k}^{-}\left(Tr\left(\boldsymbol{\sigma }\right)- \sigma _{eq}^{*-} \right) &                                                        \hbox{ if } {k}^{-}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*-} \right) \ge                                                        \dot{\vartheta }_{max}^{-}\\                                 g=\dot{\vartheta }_{max}^{-}                                                                         &                                                          \hbox{ if } {k}^{-}\left(Tr\left(\boldsymbol{\sigma }\right)- {\sigma }_{eq}^{*-} \right) <                                                          \dot{\vartheta }_{max}^{-}                                  \end{array} \right.</math>
 :<code>else</code>   (''homeostatic equilibrium'')
 :<math>g=0</math>:
@@ Line 3,459: / Line 3,459: @@
 :Calculate the residue, <math display="inline">{R}^{g}=\vartheta ^{n}-\vartheta ^{n+1}+\Delta \vartheta </math>.
 :<code>If <math display="inline">\left(\left|{R}^{g}\right|> toler.\right)</math>  then</code> (calculate slope)
-:<math display="inline">\left\{ \begin{array}{c} {ll}                                               \partial \dot{\vartheta } / \partial \vartheta = 0        &  \textrm{ if } \Delta \vartheta = 0  \\[2mm]                                              \left. \begin{array}{c} {l l}                                                      \partial \dot{\vartheta } / \partial \vartheta = -9{k}^{+}J/\vartheta ^{4}  &                                                                                     \textrm{ if } g < \dot{\vartheta }_{max}^{+} \\                                                     \partial \dot{\vartheta } / \partial \vartheta = -9 {\vartheta }_{max}^{+} J/                                                      \left(\vartheta ^{4} \left(Tr\left(\boldsymbol{\sigma }\right)-                                                     {\sigma }_{eq}^{*+} \right)\right)&                                                                                        \textrm{ if } g = \dot{\vartheta }_{max}^{+}  \\                                              \end{array}  \right\}  & \textrm{ if } \Delta \vartheta \ne 0                                           \end{array} \right.</math>
+:<math display="inline">\left\{ \begin{array}{ll}                                               \partial \dot{\vartheta } / \partial \vartheta = 0        &  \textrm{ if } \Delta \vartheta = 0  \\[2mm]                                              \left. \begin{array}{c} {l l}                                                      \partial \dot{\vartheta } / \partial \vartheta = -9{k}^{+}J/\vartheta ^{4}  &                                                                                     \hbox{ if } g < \dot{\vartheta }_{max}^{+} \\                                                     \partial \dot{\vartheta } / \partial \vartheta = -9 {\vartheta }_{max}^{+} J/                                                      \left(\vartheta ^{4} \left(Tr\left(\boldsymbol{\sigma }\right)-                                                     {\sigma }_{eq}^{*+} \right)\right)&                                                                                        \hbox{ if } g = \dot{\vartheta }_{max}^{+}  \\                                              \end{array}  \right\}  & \textrm{ if } \Delta \vartheta \ne 0                                           \end{array} \right.</math>
 :<code>end</code>
 :Update the growth stretch increment, <math display="inline">\Delta \vartheta = R^{g}\big/\left(1 -  \Delta t \;  \partial \dot{\vartheta } / \partial \vartheta \right)</math>.
@@ Line 3,487: / Line 3,487: @@
 <li>Compute the elastic part of the deformation gradient tensor, <math display="inline">\mathbf{F}^{e}= \vartheta ^{-1} \, \mathbf{F}</math>.        </li>
 <li>Calculate the second Piola-Kirchhoff stress and the corresponding tangent constitutive tensor using               the chosen constitutive model (e.g., sections&nbsp;[[#2.2.5 Neo-Hookean hyperelasticity|2.2.5]] and [[#2.2.6 Ogden hyperelasticity|2.2.6]]), <math display="inline">\mathbf{S} =\mathbf{S}\left(\mathbf{F}^{e}, J^{e},  p \right)</math> and <math display="inline"> \boldsymbol{\boldsymbol{C}}^{tan} =  \boldsymbol{\boldsymbol{C}}^{tan}\left(\mathbf{F}^{e}, J^{e},  p \right)</math>.      </li>
 </ol>
 The main characteristics of the growth model implemented in PLCd&nbsp;<span id='citeF-1'></span>[[#cite-1|[1]]] are  presented here by means of two representative three-dimensional examples.  A homogeneous state under uniaxial tension is reproduced in a single-element  problem with the aim of illustrating the basic constitutive characteristics  of the growth formulation and the role of the material parameters  of the model. Then, a rectangular plate with a double notch at its centre  and varying biological availability throughout the specimen is subjected to a tensile  load in order to show how growth is affected by biological availability.

@@ Line 3,436: / Line 3,436: @@
 where <math display="inline">\widetilde{\boldsymbol{\sigma }}</math> is the deviator part of the Cauchy  stress tensor, which is null, <math display="inline">\boldsymbol{\sigma }_{vol}</math> is its  volumetric part and <math display="inline">{p}^{av}</math> is the averaged hydrostatic elemental pressure.  This allows implementing the model in a TL framework but avoids the need to perform  a push-forward on the second Piola-Kirchhoff stress to obtain the Cauchy stress   and, then, compute its trace. It was deemed that this would introduce unnecessary numerical  noise. In addition, the second Piola-Kirchhoff stress of the present  iteration previous to including growth effects would have to be calculated,  incrementing the overall calculation time. The stress obtained in the previous iteration  could be used instead, but this also would have required  considerable changes in the code. Therefore, [[#eq-3.26|3.26]] was  considered the most effective way to obtain the trace of the Cauchy  stress tensor given the circumstances.
 {|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;
@@ Line 3,444: / Line 3,443: @@
 :Growth stretch <math>\vartheta ^{n+1}</math>=<math>\vartheta ^{n}=1</math>.
 ''Algorithm at each load increment <math>n</math>''
 :Given: deformation gradient tensor <math>\mathbf{F}</math>, elemental pressure <math>p</math>, averaged elemental pressure <math>{p}^{av}</math>,      hyperelastic material properties, growth/atrophy parameters <math>\sigma _{eq}^{*+}</math>, <math>\sigma _{eq}^{*-}</math>, <math>k^{+}</math>, <math>k^{-}</math>, <math>T_{max}^{+}</math> and <math>T_{max}^{-}</math> and biological availability parameters <math>R_{i}</math> and <math>A_{i}</math>.
 :Read value of growth stretch from previous converged step, <math>\vartheta ^{n}</math>.
 <ol>
-<li>Calculate the growth stretch increment predictor, <math display="inline">\Delta \vartheta = g\, \Delta t</math>:   </li>
+<li>Calculate the '''growth stretch increment predictor''', <math display="inline">\Delta \vartheta = g\, \Delta t</math>:   </li>
 :Compute the trace of the Cauchy stress tensor, <math display="inline">Tr\left(\boldsymbol{\sigma }\right)=-3{p}^{av}</math>.
 :Calculate the maximum growth and atrophy rates, <math display="inline">\dot{\vartheta }_{max}^{+}= T_{max}^{+} \vartheta / 3</math> and <math display="inline">\dot{\vartheta }_{max}^{-}= T_{max}^{-} \vartheta / 3</math>.
@@ Line 3,465: / Line 3,463: @@
 :Update the growth stretch increment, <math display="inline">\Delta \vartheta = R^{g}\big/\left(1 -  \Delta t \;  \partial \dot{\vartheta } / \partial \vartheta \right)</math>.
 </ol>
 |}
 {|  class="floating_tableSCP" style="text-align: left; margin: 1em auto;border-top: 2px solid;border-bottom: 2px solid;min-width:50%;"
-|+ style="font-size: 75%;" |'''Table. 8''' Algorithm at Gauss point level of the numerical integration in PLCd&nbsp;<span id='citeF-1'></span>[[#cite-1|[1]]] of the stress-driven volumetric growth model considering biological availability. The algorithm is implemented in a TL framework using hybrid elements.
+|+ style="font-size: 75%;" |'''Table. 8 (continued)''' Algorithm at Gauss point level of the numerical integration in PLCd&nbsp;<span id='citeF-1'></span>[[#cite-1|[1]]] of the stress-driven volumetric growth model considering biological availability. The algorithm is implemented in a TL framework using hybrid elements.
 |-
 | style="font-size: 75%;" |Algorithm at each load increment <math>n</math> (continued)