Meyer 2017a - Revision history

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← Older revision		Revision as of 10:46, 16 May 2017
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		+	==Abstract==
		+	In this work we compare two frameworks for thermodynamically consistent
		+	hyperelasto-plasticity with kinematic hardening. The first was formulated by Dettmer and
		+	Reese (2004), inspired by Lion (2000), and has been used to model sheet metal forming.
		+	The second, formulated by Wallin et al. (2003), has been used to model large shear
		+	strains and cyclic ratcheting behavior of pearlitic steel (Johansson et al. 2006). In this
		+	paper we show that these frameworks can result in equivalent models for certain choices
		+	of free energies. Furthermore, it is shown that the choices of free energy found in the
		+	literature only result in minor differences. These differences are discussed theoretically
		+	and investigated numerically.
		+
		+	==Full document==
	<pdf>Media:Meyer_2017a_7099_complas2017.pdf</pdf>		<pdf>Media:Meyer_2017a_7099_complas2017.pdf</pdf>
		+
		+	==References==
		+	{\|
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[1]
		+	\| style="width:20px;" \|
		+	\| F. A. M Alwahdi, A. Kapoor, and F. J. Franklin. "Subsurface microstructural analysis and mechanical properties of pearlitic rail steels in service". Wear 302.1-2 (2013), pp. 1453-1460. doi: 10.1016/j.wear.2012.12.058.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[2]
		+	\| style="width:20px;" \|
		+	\| K. Cvetkovski and J. Ahlström. "Characterisation of plastic deformation and thermal softening of the surface layer of railway passenger wheel treads". Wear 300.1-2 (2013), pp. 200-204. doi: 10.1016/j.wear.2013.01.094.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[3]
		+	\| style="width:20px;" \|
		+	\| A. Lion. "Constitutive modelling in finite thermoviscoplasticity: a physical approachbased on nonlinear rheological models". International Journal of Plasticity 16.5 (2000), pp. 469-494. doi: 10.1016/S0749-6419(99)00038-8.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[4]
		+	\| style="width:20px;" \|
		+	\| W. Dettmer and S. Reese. "On the theoretical and numerical modelling of Armstrong-Frederick kinematic hardening in the finite strain regime". In: Computer Methods in Applied Mechanics and Engineering 193.1 (2004), pp. 87-116. doi: 10.1016/j.cma.2003.09.005.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[5]
		+	\| style="width:20px;" \|
		+	\| M. Wallin, M. Ristinmaa, and N. S. Ottosen. "Kinematic hardening in large strain plasticity". In: European Journal of Mechanics - A/Solids 22.3 (2003), pp. 341-356. doi: 10.1016/S0997-7538(03)00026-3.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[6]
		+	\| style="width:20px;" \|
		+	\| M. Wallin and M. Ristinmaa. "Deformation gradient based kinematic hardening model". In: International Journal of Plasticity 21.10 (2005), pp. 2025-2050. doi:10.1016/j.ijplas.2005.01.007.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[7]
		+	\| style="width:20px;" \|
		+	\|G. Johansson, J. Ahlström, and M. Ekh. "Parameter identification and modeling of large ratcheting strains in carbon steel". In: Computers and Structures 84.15-16 (2006), pp. 1002-1011. doi: 10.1016/j.compstruc.2006.02.016.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[8]
		+	\| style="width:20px;" \|
		+	\| N. Larijani, G. Johansson, and M. Ekh. "Hybrid micro-macromechanical modeling of anisotropy evolution in pearlitic steel". In: European Journal of Mechanics - A/Solids 38 (Mar. 2013), pp. 38-47. doi: 10.1016/j.euromechsol.2012.09.011.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[9]
		+	\| style="width:20px;" \|
		+	\| J. C. Simo. "A Framework For Finite Strain Based on Maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation". In: Computer Methods in Applied Mechanics and Engineering 66 (1988), pp. 199-219.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[10]
		+	\| style="width:20px;" \|
		+	\| A. Lion. I. N. Vladimirov, M. P. Pietryga, and S. Reese. "On the modelling of non-linear kinematic hardening at finite strains with application to springback - Comparison of time integration algorithms". In: International Journal for Numerical Methods in Engineering 75.1 (July 2008), pp. 1-28. doi: 10.1002/nme.2234.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[11]
		+	\| style="width:20px;" \|
		+	\| Y. Zhu et al. "An extended cyclic plasticity model at finite deformations describing the Bauschinger effect and ratchetting behavior". In: 13th International Conference on Fracture 2013, ICF 2013 5 (2013), pp. 1-11.
		+	\|- style="vertical-align:top;"
		+	\| style="text-align:right;" \|[12]
		+	\| style="width:20px;" \|
		+	\| J. Korelc. "Multi-language and Multi-environment Generation of Nonlinear Finite Element Codes". In: Engineering with Computers 18.4 (Nov. 2002), pp. 312-327. doi: 10.1007/s003660200028.
		+	\|}

@@ Line 1: / Line 1: @@
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+<pdf>Media:Meyer_2017a_7099_complas2017.pdf</pdf>
-|-
-|<pdf>Media:Meyer_2017a_7099_complas2017.pdf</pdf>
-|- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1:''' Configurations and deformation gradients
-|}

@@ Line 1: / Line 1: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|<pdf>Media:Meyer_2017a.pdf</pdf>
+|<pdf>Media:Meyer_2017a_7099_complas2017.pdf</pdf>
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 1:''' Configurations and deformation gradients
 |}

@@ Line 31: / Line 31: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:COMPLAS_FIG1.pdf|600px|Configurations and deformation gradients]]
+|<pdf>Media:Meyer_2017a_3352_Figure_1.pdf</pdf>
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 1:''' Configurations and deformation gradients

@@ Line 465: / Line 465: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:draft_Meyer_496742701-picture-e57868.png|600px|Uniaxial response]]
+|[[File:COMPLAS_Fig2a.pdf|500px|Uniaxial response]]
-|[[Image:draft_Meyer_496742701-picture-579ecd.png|400px|Simple shear response]]
+|[[Image:COMPLAS_Fig2b.pdf|500px|Simple shear response]]
 |- style="text-align: center; font-size: 75%;"
 | (a) Uniaxial response

@@ Line 513: / Line 513: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="width:300px; text-align: right;" | <math>\bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}} =</math>
+| style="width:270px; text-align: right;" | <math>\bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}} =</math>
 | style="text-align: left;" | <math> \dot{\lambda }\frac{   {}\mathrm{^1}\bar{\bar{\boldsymbol{M}}}^\mathrm{t}_{\mathrm{k}}\mathrm{}}{b_\infty } </math>
 |}
@@ Line 526: / Line 526: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="width:300px; text-align: right;" | <math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= </math>
+| style="width:270px; text-align: right;" | <math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= </math>
 | style="text-align: left;" | <math> \bar{\boldsymbol{L}}_{\mathrm{p}}- \dot{\lambda }\frac{   {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\mathrm{}}{b_\infty } </math>
 |}
@@ Line 538: / Line 538: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="width:300px; text-align: right;" | <math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= </math>
+| style="width:270px; text-align: right;" | <math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= </math>
 | style="text-align: left;" | <math>\dot{\lambda }\frac{\partial \Phi ^*_{\mathrm{kin}}}{\partial \left(   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\right)} = -\bar{\boldsymbol{L}}_{\mathrm{p}}+ \dot{\lambda }\frac{   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\mathrm{}}{b_\infty } </math>
 |}

@@ Line 202: / Line 202: @@
 |-
 | style="text-align: center;" |
-<math>    {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}= \frac{\partial }{\partial \boldsymbol{F}_{\mathrm{ke}}}\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} </math>
+<math>    {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}= \frac{\partial \Psi_\mathrm{kin}}{\partial \boldsymbol{F}_{\mathrm{ke}}}\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (13)
 |}
@@ Line 219: / Line 219: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}= -\frac{\partial }{\partial \boldsymbol{F}_{\mathrm{ke}}}\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} </math>
+| style="text-align: center;" | <math>{}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}= -\frac{\partial \Psi_\mathrm{kin}}{\partial \boldsymbol{F}_{\mathrm{ke}}}\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (14)
@@ Line 333: / Line 333: @@
 |-
 | style="width:150px; text-align: right;" | <math>{}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}=</math>
-| style="text-align: left;" | <math>\left(\frac{\partial }{\partial \bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}}:\frac{\partial }{\partial \boldsymbol{F}_{\mathrm{ke}}}\right)\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} = 2\boldsymbol{F}_{\mathrm{ke}}\left(\frac{\partial }{\partial \bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}} \right)\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}}  </math>
+| style="text-align: left;" | <math>\left(\frac{\partial \Psi_\mathrm{kin}}{\partial \bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}}:\frac{\partial \bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}}{\partial \boldsymbol{F}_{\mathrm{ke}}}\right)\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} = 2\boldsymbol{F}_{\mathrm{ke}}\left(\frac{\partial \Psi_\mathrm{kin}}{\partial \bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}} \right)\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}}  </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (24)
 |-
 | style="width:150px; text-align: right;" | <math>   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}=</math>
-| style="text-align: left;" | <math>-\left(\frac{\partial }{\partial \bar{\boldsymbol{c}}_{\mathrm{ke}}}:\frac{\partial }{\partial \boldsymbol{F}_{\mathrm{ke}}}\right)\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} = 2\bar{\boldsymbol{c}}_{\mathrm{ke}}\frac{\partial }{\partial \bar{\boldsymbol{c}}_{\mathrm{ke}}}  </math>
+| style="text-align: left;" | <math>-\left(\frac{\partial \Psi_\mathrm{kin}}{\partial \bar{\boldsymbol{c}}_{\mathrm{ke}}}:\frac{\partial \bar{\boldsymbol{c}}_{\mathrm{ke}}}{\partial \boldsymbol{F}_{\mathrm{ke}}}\right)\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}} = 2\bar{\boldsymbol{c}}_{\mathrm{ke}}\frac{\partial \Psi_\mathrm{kin}}{\partial \bar{\boldsymbol{c}}_{\mathrm{ke}}}  </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (25)
 |}
@@ Line 539: / Line 539: @@
 |-
 | style="width:200px; text-align: right;" | <math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= </math>
-| style="text-align: left;" | <math>\dot{\lambda }\frac{\partial }{\partial \left(   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\right)} = -\bar{\boldsymbol{L}}_{\mathrm{p}}+ \dot{\lambda }\frac{   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\mathrm{}}{b_\infty } </math>
+| style="text-align: left;" | <math>\dot{\lambda }\frac{\partial \Phi ^*_{\mathrm{kin}}}{\partial \left(   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\right)} = -\bar{\boldsymbol{L}}_{\mathrm{p}}+ \dot{\lambda }\frac{   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\mathrm{}}{b_\infty } </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (38)

@@ Line 120: / Line 120: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>D = \boldsymbol{\tau }:\boldsymbol{l} - \dot{\Psi } =  \boldsymbol{\tau }:\boldsymbol{l} - \frac{\partial }{\partial \bar{\boldsymbol{C}}_{\mathrm{e}}}\dot{\bar{\boldsymbol{C}}}_{\mathrm{e}}- \frac{\partial }{\partial \boldsymbol{F}_{\mathrm{ke}}}:\dot{\boldsymbol{F}}_{\mathrm{ke}}\geq 0 </math>
+| style="text-align: center;" | <math>D = \boldsymbol{\tau }:\boldsymbol{l} - \dot{\Psi } =  \boldsymbol{\tau }:\boldsymbol{l} - \frac{\partial \Psi_\mathrm{e}}{\partial \bar{\boldsymbol{C}}_{\mathrm{e}}}\dot{\bar{\boldsymbol{C}}}_{\mathrm{e}}- \frac{\partial \Psi_\mathrm{kin} }{\partial \boldsymbol{F}_{\mathrm{ke}}}:\dot{\boldsymbol{F}}_{\mathrm{ke}}\geq 0 </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (7)
@@ Line 133: / Line 133: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>D = \bar{\boldsymbol{M}}:\bar{\boldsymbol{L}}_{\mathrm{p}}- \left(\frac{\partial }{\partial \boldsymbol{F}_{\mathrm{ke}}}\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}}\right):\bar{\boldsymbol{L}}_{\mathrm{ke}} </math>
+| style="text-align: center;" | <math>D = \bar{\boldsymbol{M}}:\bar{\boldsymbol{L}}_{\mathrm{p}}- \left(\frac{\partial \Psi_\mathrm{kin}}{\partial \boldsymbol{F}_{\mathrm{ke}}}\boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}}\right):\bar{\boldsymbol{L}}_{\mathrm{ke}} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (8)
@@ Line 146: / Line 146: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\bar{\boldsymbol{M}} =  2\bar{\boldsymbol{C}}_{\mathrm{e}}\frac{\partial }{\partial \bar{\boldsymbol{C}}_{\mathrm{e}}}  </math>
+| style="text-align: center;" | <math>\bar{\boldsymbol{M}} =  2\bar{\boldsymbol{C}}_{\mathrm{e}}\frac{\partial \Psi_\mathrm{e}}{\partial \bar{\boldsymbol{C}}_{\mathrm{e}}}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (9)
@@ Line 159: / Line 159: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\bar{\boldsymbol{L}}_{\mathrm{p}}= \dot{\lambda }\frac{\partial }{\partial \bar{\boldsymbol{M}}}  </math>
+| style="text-align: center;" | <math>\bar{\boldsymbol{L}}_{\mathrm{p}}= \dot{\lambda }\frac{\partial \Phi}{\partial \bar{\boldsymbol{M}}}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)