A COMPARISON OF TWO FRAMEWORKS FOR KINEMATIC HARDENING IN HYPERELASTO-PLASTICITY

==A Comparison of two frameworks for kinematic hardening in hyperelasto-plasticity==

'''Knut A. Meyer<math>^{*}</math>, Magnus Ekh'''

{|
|-
|Department of Applied Mechanics, Division of Material and Computational Mechanics, 
|-
| Chalmers University of Technology, 412 96 Gothenburg, Sweden
|-
| e-mail<math>^{*}</math>: [mailto:knut.andreas.meyer@chalmers.se knut.andreas.meyer@chalmers.se], web page: [http://www.chalmers.se/ http://www.chalmers.se/]
|}

==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.

'''keywords''' Hyperelasto-plasticity, Finite strains, Kinematic hardening



==1 INTRODUCTION==

Large strains in metals during room temperature occur in many technical applications, often during manufacturing, such as sheet metal forming. Some components are also subjected to large strains during service, for example in the surface layer of railway rails and wheels (see e.g. [1, 2]). Experiments have shown that the Bauchinger effect, often modeled with kinematic hardening, is pronounced in many metals. Kinematic hardening can be modeled with different thermodynamically consistent hyperelasto-plastic frameworks found in the literature, and two of them are considered here. The first framework is based on rheological models with an Armstrong-Frederick (AF) type of kinematic hardening, and was proposed by Lion [3] and further developed by Dettmer and Reese [4]. The second framework, introduced by Wallin et al. [5], also features an AF type of kinematic hardening, and has been used to model the Swift effect [6] and large deformations in railway applications [7, 8]. In this paper we compare these frameworks, both theoretically and numerically.

==2 DESCRIPTION OF FRAMEWORKS==

In this section, the modeling frameworks are presented in terms of their assumed kinematics and thermodynamics. The common parts are presented in Subsections 2.1 and 2.2, followed by a description of how the frameworks differ in Subsections 2.3 and 2.4. Three specific models are defined in Subsections 2.5 and 2.6, which are compared in the numerical examples in Section 3. For the clarity of the presentation, only linear kinematic hardening is considered in the current section. In Section 4 we investigate nonlinear kinematic hardening, e.g. of Armstrong-Frederick type.

<div id='img-1'></div>
{| 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]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Configurations and deformation gradients
|}

===2.1 Kinematics and notations===

Figure 1 shows the different configurations used in both [4] and [5]. Dettmer and Reese [4] introduce the inelastic plastic deformation gradient <math display="inline">\boldsymbol{F}_{\mathrm{kp}}</math> connecting the fictitious kinematic configuration <math display="inline">\bar{\bar{\Omega }}</math> to the initial configuration <math display="inline">\Omega </math>. This connection is not introduced in Wallin et al. [5], but otherwise the same configurations and remaining deformation gradients are present in both frameworks.

Tensors on the current configuration <math display="inline">\omega </math> are denoted with lower case letters and no bars, e.g. <math display="inline">\boldsymbol{b}</math>. Tensors on the intermediate <math display="inline">\bar{\Omega }</math> and kinematic <math display="inline">\bar{\bar{\Omega }}</math> configurations are denoted by one bar, e.g. <math display="inline">\bar{\boldsymbol{C}}_{\mathrm{e}}</math>, and two bars, e.g. <math display="inline">\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}</math>, respectively. The following decompositions of the deformation gradients and definitions of the deformation tensors will be used:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{rlrlrl}\boldsymbol{F} &= \boldsymbol{F}_{\mathrm{e}}\boldsymbol{F}_{\mathrm{p}} \quad  &\bar{\boldsymbol{C}}_{\mathrm{e}}&= \boldsymbol{F}^{\mathrm{t}}_{\mathrm{e}}\boldsymbol{F}_{\mathrm{e}} \quad  &\boldsymbol{c}_{\mathrm{e}}&= \boldsymbol{F}^{-\mathrm{t}}_{\mathrm{e}}\boldsymbol{F}^{-1}_{\mathrm{e}}= \boldsymbol{b}^{-1}_{\mathrm{e}} \\[1mm] \boldsymbol{F}_{\mathrm{p}}&= \boldsymbol{F}_{\mathrm{ke}}\boldsymbol{F}_{\mathrm{kp}} \quad  &\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}&= \boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}}\boldsymbol{F}_{\mathrm{ke}} \quad  &\bar{\boldsymbol{c}}_{\mathrm{ke}}&= \boldsymbol{F}^{-\mathrm{t}}_{\mathrm{ke}}\boldsymbol{F}^{-1}_{\mathrm{ke}}= \bar{\boldsymbol{b}}_{\mathrm{ke}}^{-1} \end{array} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

The velocity gradients on the intermediate and kinematic configurations are defined as

<span id="eq-2"></span>
<span id="eq-3"></span>
<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol{L}}_{\mathrm{p}}= \dot{\boldsymbol{F}}_{\mathrm{p}}\boldsymbol{F}^{-1}_{\mathrm{p}}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|-
| style="text-align: center;" | <math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= \dot{\boldsymbol{F}}_{\mathrm{ke}}\boldsymbol{F}^{-1}_{\mathrm{ke}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|-
| style="text-align: center;" | <math> \bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}}= \dot{\boldsymbol{F}}_{\mathrm{kp}}\boldsymbol{F}^{-1}_{\mathrm{kp}}   </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}
|}

and hence the velocity gradient <math display="inline">\boldsymbol{l}</math> on the current configuration can be written as

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\boldsymbol{l} = \dot{\boldsymbol{F}}\boldsymbol{F}^{-1}=\dot{\boldsymbol{F}}_{\mathrm{e}}\boldsymbol{F}^{-1}_{\mathrm{e}}+ \boldsymbol{F}_{\mathrm{e}}\bar{\boldsymbol{L}}_{\mathrm{p}}\boldsymbol{F}^{-1}_{\mathrm{e}}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

===2.2 Thermodynamics===

The free energy <math display="inline">\Psi </math> is introduced with the additive split according to

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Psi = \Psi _{\mathrm{e}}\left(\bar{\boldsymbol{C}}_{\mathrm{e}}\right)+ \Psi _{\mathrm{kin}}\left(\boldsymbol{F}_{\mathrm{ke}}\right) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

whereby the dissipation inequality (see e.g. Simo (1998) [9]) becomes

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| 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="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

where <math display="inline">\boldsymbol{\tau }</math> is the Kirchhoff stress.  Using the requirement of zero dissipation during elastic loading and Equation [[#eq-5|5]], the reduced dissipation inequality becomes

<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| 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="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

where the Mandel stress <math display="inline">\bar{\boldsymbol{M}}</math> is defined as

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| 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="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

In a standard fashion, we adopt an associative evolution of the plastic deformation gradient in this paper:

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| 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="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

where the functional dependence of the yield function <math display="inline">\Phi \leq 0</math> will be specified later. Up until this point the two frameworks are identical. We first describe the framework by Dettmer and Reese [4], before proceeding with the framework by Wallin et al. [5].

===2.3 1st framework [4]===

In the first framework proposed by Lion [3] and further developed by Dettmer and Reese [4], the plastic deformation gradient <math display="inline">\boldsymbol{F}_{\mathrm{p}}</math> is multiplicatively decomposed into an elastic part <math display="inline">\boldsymbol{F}_{\mathrm{ke}}</math> and a plastic part <math display="inline">\boldsymbol{F}_{\mathrm{kp}}</math>. The physical motivation is that <math display="inline">\boldsymbol{F}_{\mathrm{ke}}</math> represents local elastic deformations on the microscale caused by dislocations and <math display="inline">\boldsymbol{F}_{\mathrm{kp}}</math> represents irreversible displacements in the slip systems. The assumption is that development of <math display="inline">\boldsymbol{F}_{\mathrm{ke}}</math> results in linear kinematic hardening and the development of <math display="inline">\boldsymbol{F}_{\mathrm{kp}}</math> reduces <math display="inline">\boldsymbol{F}_{\mathrm{ke}}</math>, hence causing saturation (dynamic recovery) of the kinematic hardening. This is illustrated using a rheological model by Lion [3].  The multiplicative split of <math display="inline">\boldsymbol{F}_{\mathrm{p}}</math> results in the following additive split of the plastic velocity gradient <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{p}}</math>

<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol{L}}_{\mathrm{p}}= \bar{\boldsymbol{L}}_{\mathrm{ke}}+ \boldsymbol{F}_{\mathrm{ke}}\bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}}\boldsymbol{F}^{-1}_{\mathrm{ke}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

For the case of purely linear hardening (<math display="inline">\boldsymbol{F}_{\mathrm{kp}}=\boldsymbol{I}</math>), the reduced dissipation, Equation [[#eq-8|8]], can be written as

<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>D = \left(\bar{\boldsymbol{M}}-    {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\right):\bar{\boldsymbol{L}}_{\mathrm{p}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

where the kinematic hardening stress of Mandel type (also denoted back-stress) is defined as
<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| 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>
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}
|}

This motivates that the driving force for plastic flow is <math display="inline">\bar{\boldsymbol{M}}-   {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math> and thereby a yield criterion expressed as <math display="inline">\Phi (\bar{\boldsymbol{M}}-   {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{})</math>.

===2.4 2nd framework [5]===

In the second framework, proposed by Wallin et al. [5], the deformation gradient <math display="inline">\boldsymbol{F}^{-1}_{\mathrm{ke}}</math> is introduced to model the deformation of the crystal lattice, due to the residual micro stresses responsible for the Bauchinger effect.  From this deformation gradient the kinematic hardening stress of Mandel type is defined as

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| 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="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

which yields that the reduced dissipation inequality [[#eq-8|8]] is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>D = \bar{\boldsymbol{M}}:\bar{\boldsymbol{L}}_{\mathrm{p}} +    {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}:\bar{\boldsymbol{L}}_{\mathrm{ke}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

Using the standard interpretation of <math display="inline">   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math> as a back-stress that reduces the driving force for plasticity, motivates the yield function <math display="inline">\Phi (\bar{\boldsymbol{M}}-   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{})</math>. This gives, by the postulate of maximum dissipation, the kinematic relation

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol{L}}_{\mathrm{p}}= - \bar{\boldsymbol{L}}_{\mathrm{ke}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}

and the same reduced dissipation inequality as in Equation [[#eq-12|12]] is obtained:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>D = \left(\bar{\boldsymbol{M}}-    {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\right):\bar{\boldsymbol{L}}_{\mathrm{p}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|}

===2.5 Specific formats for free energy===

The elastic and kinematic free energies (with the third invariant <math display="inline">I_{\mathrm{3\bullet }} = \det (\bullet )</math>) proposed by Vladimirov et al [10] are

<span id="eq-18"></span>
<span id="eq-19"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: right;" | <math>{}\mathrm{^A}\Psi _{\mathrm{e}}\mathrm{} =</math>
| style="text-align: left;" | <math>\frac{1}{2}G\left(\mathrm{tr}(\bar{\boldsymbol{C}}_{\mathrm{e}}) - 3 - \ln \left(I_{\mathrm{3C}_{\mathrm{e}}}\right)\right) + \frac{\Lambda }{4}\left(I_{\mathrm{3C}_{\mathrm{e}}} - 1 - \ln \left(I_{\mathrm{3C}_{\mathrm{e}}}\right)\right)  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|-
| style="text-align: right;" | <math>    {}\mathrm{^A}\Psi _{\mathrm{kin}}\mathrm{} =</math>
| style="text-align: left;" | <math>\frac{1}{2}H_{\mathrm{kin}}\left(\mathrm{tr}(\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}) - 3 - \ln \left(I_{\mathrm{3C}_{\mathrm{ke}}}\right)\right)  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
|}
|}

The part of the elastic free energy, corresponding to Lamé's second parameter <math display="inline">\Lambda </math>, is thus not included in the kinematic free energy.

A similar split is introduced in [5], but the free energy in that work is decomposed into an isochoric and a volumetric part. The formulation for the volumetric part is not the same in [5] and [7, 8], and here we use the formulation from [7, 8]. This difference only affects the bulk elastic response and the influence on the numerical results studied in this paper is therefore negligible.

<span id="eq-20"></span>
<span id="eq-21"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: right;" | <math> {}\mathrm{^B}\Psi _{\mathrm{e}}\mathrm{} = </math>
| style="text-align:left;" | <math>\frac{1}{2}G\left(\mathrm{tr}\left(I_{\mathrm{3C}_{\mathrm{e}}}^{-1/3} \bar{\boldsymbol{C}}_{\mathrm{e}}\right)- 3\right)+ \frac{1}{2}K\left(I_{\mathrm{3C}_{\mathrm{e}}}^{1/2} - 1 \right)^2  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
|-
| style="text-align: right;" | <math>    {}\mathrm{^B}\Psi _{\mathrm{kin}}\mathrm{} = </math>
| style="text-align:left;"| <math>\frac{1}{2}H_{\mathrm{kin}}\left(\mathrm{tr}\left(I_{\mathrm{3c}_{\mathrm{ke}}}^{-1/3} \bar{\boldsymbol{c}}_{\mathrm{ke}}\right)- 3\right)  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
|}
|}

From the discussion so far, there seem to be several differences between the frameworks: (1) the definition of the Mandel back-stress (<math display="inline">   {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math> or <math display="inline">   {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math>), (2) the variable of which the kinematic free energy depends on (<math display="inline">\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}</math> or <math display="inline">\bar{\boldsymbol{c}}_{\mathrm{ke}}</math>) and (3) what part of the elastic free energy formulation that is used to formulate the kinematic free energy. The third of these can be investigated by taking the format of free energy from the second framework, but using the definitions and variables from the first framework to obtain model <math display="inline">\mathrm{C}</math>:

<span id="eq-22"></span>
<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: right;" | <math>{}\mathrm{^C}\Psi _{\mathrm{e}}\mathrm{} =</math>
| style="text-align: left;" | <math>{}\mathrm{^B}\Psi _{\mathrm{e}}\mathrm{}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
|-
| style="text-align: right;" | <math>    {}\mathrm{^C}\Psi _{\mathrm{kin}}\mathrm{} =</math>
| style="text-align: left;" | <math>\frac{1}{2}H_{\mathrm{kin}}\left(\mathrm{tr}\left(I_{\mathrm{3C}_{\mathrm{ke}}}^{-1/3} \bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}\right)- 3\right)  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
|}
|}

===2.6 Stresses for each model===

We have now described both frameworks, the first by Dettmer and Reese [4] and the second by Wallin et al. [5]. By letting <math display="inline">\Psi _{\mathrm{kin}}</math> depend on <math display="inline">\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}</math> or <math display="inline">\bar{\boldsymbol{c}}_{\mathrm{ke}}</math> we can use [[#eq-13|13]] and [[#eq-14|14]], respectively, to obtain the Mandel back-stresses for the two frameworks:

<span id="eq-24"></span>
<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}= \quad \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="width: 5px;text-align: right;white-space: nowrap;" | (24)
|-
| style="text-align: center;" | <math>    {}\mathrm{^2}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}= -\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="width: 5px;text-align: right;white-space: nowrap;" | (25)
|}
|}

The Mandel stresses for model <math display="inline">\mathrm{A}</math>, <math display="inline">   {}\mathrm{^A}\bar{\boldsymbol{M}}\mathrm{}</math> and <math display="inline">   {}\mathrm{^A}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math>, are found using Equations [[#eq-9|9]] and [[#eq-24|24]] with the free energies in Equations [[#eq-18|18]] and [[#eq-19|19]]:

<span id="eq-26"></span>
<span id="eq-27"></span>
<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}\mathrm{^A}\bar{\boldsymbol{M}}\mathrm{} = G (\bar{\boldsymbol{C}}_{\mathrm{e}}- \boldsymbol{I}) + \frac{\Lambda }{2} (I_\mathrm{3C_e} - 1)\boldsymbol{I}, \quad     {}\mathrm{^A}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{} = H_{\mathrm{kin}}\left(\bar{\boldsymbol{b}}_{\mathrm{ke}}-\boldsymbol{I}\right)  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
|-
| style="text-align: center;" |

The stresses for model <math>\mathrm{B}</math>, <math>   {}\mathrm{^B}\bar{\boldsymbol{M}}\mathrm{}</math> and <math>   {}\mathrm{^B}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math>, are given by using Equations [[#eq-9|9]] and [[#eq-25|25]] with the free energies in Equations [[#eq-20|20]] and [[#eq-21|21]]:

<math>    {}\mathrm{^B}\bar{\boldsymbol{M}}\mathrm{} = G I_\mathrm{3C_e}^{-1/3} \bar{\boldsymbol{C}}_{\mathrm{e}}^{\mathrm{dev}}+ K(I_\mathrm{3C_e} - I_\mathrm{3C_e}^{1/2})\boldsymbol{I}, \quad     {}\mathrm{^B}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{} = H_{\mathrm{kin}}I_{\mathrm{c}_{\mathrm{ke}}}^{-1/3} \bar{\boldsymbol{c}}_{\mathrm{ke}}^{\mathrm{dev}}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
|-
| style="text-align: center;" |

For model <math>\mathrm{C}</math>, we use the first framework, i.e. the stresses <math>   {}\mathrm{^C}\bar{\boldsymbol{M}}\mathrm{}</math> and <math>   {}\mathrm{^C}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}</math> are given by using Equations [[#eq-9|9]] and [[#eq-24|24]], but with the free energies in Equations [[#eq-22|22]] and [[#eq-23|23]]. We further note that <math>I_{3\mathrm{C}_{\mathrm{ke}}}=I_{3\mathrm{b}_{\mathrm{ke}}}</math> and <math>\mathrm{tr}(\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}})=\mathrm{tr}(\bar{\boldsymbol{b}}_{\mathrm{ke}})</math>, which leads to:

<math>    {}\mathrm{^C}\bar{\boldsymbol{M}}\mathrm{} =    {}\mathrm{^B}\bar{\boldsymbol{M}}\mathrm{} \quad     {}\mathrm{^C}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{} = H_{\mathrm{kin}}I_{\mathrm{b}_{\mathrm{ke}}}^{-1/3} \bar{\boldsymbol{b}}_{\mathrm{ke}}^{\mathrm{dev}}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
|}

If <math display="inline">   {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{} =    {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{}</math> then clearly model <math display="inline">\mathrm{B}</math> and <math display="inline">\mathrm{C}</math> are equivalent. Assuming that is the case for some point in time, we also have <math display="inline">   {}\mathrm{^B}\bar{\boldsymbol{L}}_{\mathrm{p}}\mathrm{}=   {}\mathrm{^C}\bar{\boldsymbol{L}}_{\mathrm{p}}\mathrm{}=\bar{\boldsymbol{L}}_{\mathrm{p}}</math>. Since model <math display="inline">\mathrm{B}</math> is using the second framework, and model <math display="inline">\mathrm{C}</math> is using the first, we also have <math display="inline">-   {}\mathrm{^B}\bar{\boldsymbol{L}}_{\mathrm{ke}}\mathrm{}=   {}\mathrm{^C}\bar{\boldsymbol{L}}_{\mathrm{ke}}\mathrm{}=\bar{\boldsymbol{L}}_{\mathrm{p}}</math>, hence

<span id="eq-29"></span>
<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}\mathrm{^B}\dot{\bar{\boldsymbol{c}}}_{\mathrm{ke}}\mathrm{} = -\left(   {}\mathrm{^B}\bar{\boldsymbol{L}}\mathrm{_{\mathrm{ke}}^{\mathrm{t}}}   {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{} +    {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{}   {}\mathrm{^B}\bar{\boldsymbol{L}}_{\mathrm{ke}}\mathrm{}\right) = \bar{\boldsymbol{L}}_{\mathrm{p}}^{\mathrm{t}}   {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{} +    {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{}\bar{\boldsymbol{L}}_{\mathrm{p}}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
|-
| style="text-align: center;" | <math>    {}\mathrm{^C}\dot{\bar{\boldsymbol{b}}}_{\mathrm{ke}}\mathrm{} = \quad \left(   {}\mathrm{^C}\bar{\boldsymbol{L}}_{\mathrm{ke}}\mathrm{}   {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{} +    {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{}   {}\mathrm{^C}\bar{\boldsymbol{L}}\mathrm{_{\mathrm{ke}}^{\mathrm{t}}}\right)= \bar{\boldsymbol{L}}_{\mathrm{p}}   {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{} +    {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{}\bar{\boldsymbol{L}}_{\mathrm{p}}^{\mathrm{t}}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
|}
|}

As <math display="inline">   {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{} =    {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{} = \boldsymbol{I}</math> initially, the statement <math display="inline">   {}\mathrm{^B}\bar{\boldsymbol{c}}_{\mathrm{ke}}\mathrm{} =    {}\mathrm{^C}\bar{\boldsymbol{b}}_{\mathrm{ke}}\mathrm{}</math> is true for all points in time under the assumption that <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{p}}</math> is symmetric. If the free energy is isotropic, the Mandel stresses <math display="inline">\bar{\boldsymbol{M}}</math> and <math display="inline">\bar{\boldsymbol{M}}_{\mathrm{k}}</math> are symmetric, and hence <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{p}}</math> becomes symmetric for the associative choice of <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{p}}</math> in Equation [[#eq-10|10]]. This leads to the conclusion that model B and C are equivalent, which is verified numerically later. Furthermore, this proof leads to the interesting conclusion that the frameworks can give exactly the same model with proper choices of free energy. The possibility to formulate an unsymmetric <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{p}}</math> for isotropy is discussed in e.g. Wallin et al. [5] and Wallin and Ristinmaa [6], but is not investigated in this paper.

==3 NUMERICAL RESULTS==

In this section we evaluate the response of the material models for uniaxial loading and simple shear loading. The models are implemented using a standard Backward Euler integration scheme, for which Vladimirov et al. [10] noted that the accuracy suffers at large time steps. To avoid these accuracy problems, approximately <math display="inline">4\cdot{10}^4</math> and <math display="inline">5\cdot 10^4</math> load steps are used for the uniaxial and simple shear loading, respectively.

The von Mises effective stress is used to define the yield function <math display="inline">\Phi </math> according to

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Phi = \sqrt{\frac{3}{2}}\sqrt{\mathrm{dev}\left(\bar{\boldsymbol{M}}^\mathrm{t}-\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\right):\mathrm{dev}\left(\bar{\boldsymbol{M}}-\bar{\boldsymbol{M}}_{\mathrm{k}}\right)} - Y_0 = f\left(\bar{\boldsymbol{M}}-\bar{\boldsymbol{M}}_{\mathrm{k}}\right)- Y_0 \leq 0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
|}

whereby the evolution of the plastic deformation gradient in [[#eq-10|10]] becomes

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\boldsymbol{L}}_{\mathrm{p}}= \dot{\lambda }\frac{3}{2} \frac{\mathrm{dev}\left(\bar{\boldsymbol{M}}^\mathrm{t}-\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\right)}{f\left(\bar{\boldsymbol{M}}-\bar{\boldsymbol{M}}_{\mathrm{k}}\right)} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
|}

From this it follows that the plastic deformation is isochoric: By time differentiation of <math display="inline">\det \left(\boldsymbol{F}_{\mathrm{p}}\right)</math> and using [[#eq-2|2]]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial }{\partial t} \det \left(\boldsymbol{F}_{\mathrm{p}}\right) = \det \left(\boldsymbol{F}_{\mathrm{p}}\right)\boldsymbol{F}^{-\mathrm{t}}_{\mathrm{p}}:\left(\bar{\boldsymbol{L}}_{\mathrm{p}}\boldsymbol{F}_{\mathrm{p}}\right) = \det \left(\boldsymbol{F}_{\mathrm{p}}\right)\mathrm{tr}\left(\bar{\boldsymbol{L}}_{\mathrm{p}}\right)= 0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33)
|}

hence, <math display="inline">\det \left(\boldsymbol{F}_{\mathrm{p}}\right)= 1</math>.

The following material parameters: <math display="inline">G=81\,\mathrm{GPa}</math>, <math display="inline">K=174\,\mathrm{GPa}</math>, <math display="inline">\Lambda = K - 2G/3</math>, <math display="inline">Y_0=100\,\mathrm{MPa}</math> and <math display="inline">H_{\mathrm{kin}}=1000\,\mathrm{MPa}</math>, are used in the numerical examples in Figure [[#img-2|2]].  <div id='img-2a'></div>
<div id='img-2b'></div>
<div id='img-2'></div>
{| 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]]
|[[Image:draft_Meyer_496742701-picture-579ecd.png|400px|Simple shear response]]
|- style="text-align: center; font-size: 75%;"
| (a) Uniaxial response
| (b) Simple shear response
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 2:''' Numerical results
|}

We first consider uniaxial stress in Figure [[#img-2a|2a]], by letting the normal deformation gradient <math display="inline">F_{11}</math> increase from 1 to 10 while keeping the Cauchy stresses <math display="inline">\sigma _{22}=\sigma _{33}=0</math> (<math display="inline">\boldsymbol{\sigma }=\boldsymbol{\tau }/I_{3\mathrm{F}}</math>). As previously shown theoretically, model <math display="inline">\mathrm{B}</math> and <math display="inline">\mathrm{C}</math> give the same response. Model <math display="inline">\mathrm{A}</math> gives a somewhat stiffer response at large deformations, due to the different choice of free energy.

In the case of simple shear loading, Figure [[#img-2b|2b]] shows that the response of all the models coincide. While this is expected for model <math display="inline">\mathrm{B}</math> and <math display="inline">\mathrm{C}</math>, the fact that model <math display="inline">\mathrm{A}</math> and <math display="inline">\mathrm{C}</math> coincide is explained with simple shear being an isochoric process (<math display="inline">I_{3\mathrm{F}}=I_{3\mathrm{F}_{\mathrm{p}}}=1 \Rightarrow I_{3\mathrm{F}_{\mathrm{e}}}=1</math>)

The results in Figure [[#img-2|2]] show negligible differences between the different formulations of free energy. From a theoretical point of view, one could argue that the deviatoric dependence of the back-stress is more correct, based on the experimental evidence of volume preservation for metal plasticity.

==4 ARMSTRONG-FREDERICK SATURATION==

Linear kinematic hardening was considered in Section [[#2 DESCRIPTION OF FRAMEWORKS|2]], for which Dettmer and Reese [4] set <math display="inline">\boldsymbol{F}_{\mathrm{kp}}=\boldsymbol{I}</math>, yielding the reduced dissipation inequality in Equation [[#eq-12|12]]. If the general case with an evolving <math display="inline">\boldsymbol{F}_{\mathrm{kp}}</math> is considered, the reduced dissipation inequality, using Equation [[#eq-11|11]], becomes

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>D = \left(\bar{\boldsymbol{M}}-    {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\right):\bar{\boldsymbol{L}}_{\mathrm{p}}+    {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}:\left(\boldsymbol{F}_{\mathrm{ke}}\bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}}\boldsymbol{F}^{-1}_{\mathrm{ke}}\right) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
|}

From this the kinematic stress of Mandel type on the kinematic configuration <math display="inline">   {}\mathrm{^1}\bar{\bar{\boldsymbol{M}}}_{\mathrm{k}}\mathrm{}</math> is defined as

<span id="eq-35"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}\mathrm{^1}\bar{\bar{\boldsymbol{M}}}_{\mathrm{k}}\mathrm{} = \boldsymbol{F}^{\mathrm{t}}_{\mathrm{ke}}   {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}\mathrm{}\boldsymbol{F}^{-\mathrm{t}}_{\mathrm{ke}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (35)
|}

Saturation is motivated by the rheological model, setting the evolution on <math display="inline">\bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}}^{\mathrm{sym}}</math>. As the Mandel stresses in [4] are symmetric, this can be written as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\bar{\boldsymbol{L}}}_{\mathrm{kp}}= \dot{\lambda }\frac{   {}\mathrm{^1}\bar{\bar{\boldsymbol{M}}}^\mathrm{t}_{\mathrm{k}}\mathrm{}}{b_\infty } </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (36)
|-
| style="text-align: center;" |

where <math>b_\infty </math> is a material parameter controlling the kinematic saturation. Equation [[#eq-11|11]] and [[#eq-35|35]] then yield

<math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= \bar{\boldsymbol{L}}_{\mathrm{p}}- \dot{\lambda }\frac{   {}\mathrm{^1}\bar{\boldsymbol{M}}_{\mathrm{k}}^\mathrm{t}\mathrm{}}{b_\infty } </math>

This Equation can be compared with Wallin et al. [5], who use a modified potential <math>\Phi ^*_{\mathrm{kin}}</math> to obtain

<math> \bar{\boldsymbol{L}}_{\mathrm{ke}}= \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="width: 5px;text-align: right;white-space: nowrap;" | (37)
|}

<math display="inline">\bar{\boldsymbol{L}}_{\mathrm{ke}}</math> will be symmetric if a modified yield potential <math display="inline">\Phi ^*_{\mathrm{kin}}</math> exists and we have, as before, symmetric Mandel stresses. This is the case for the considered model with Armstrong-Frederick type of non-linear kinematic saturation. <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{ke}}</math> is also symmetric in the work by Zhu et al. [11], where the first framework was extended to include nonlinear kinematic hardening of Ohno-Wang type. When <math display="inline">\bar{\boldsymbol{L}}_{\mathrm{ke}}</math> is symmetric, the same arguments as before relating to Equations [[#eq-29|29]] and [[#eq-30|30]] hold true. Hence, for appropriate free energies the two frameworks give the same model also for nonlinear kinematic hardening.

==5 CONCLUDING REMARKS==

We have shown that the two different frameworks, introduced by [4] and [5], can give equivalent models for isotropic free energies. The major difference between the models used for the different frameworks is the kinematic free energy. To obtain the same model, the same structure of the kinematic free energy must be used, but with a different variable (<math display="inline">\bar{\bar{\boldsymbol{C}}}_{\mathrm{ke}}</math> or <math display="inline">\bar{\boldsymbol{c}}_{\mathrm{ke}}</math>). The numerical results confirm these theoretical findings. They further show that the difference between the formulations of free energy has a negligible effect on the material response up to a stretch of 5 for uniaxial loading, and no effect during simple shear.



==6 ACKNOWLEDGMENTS==

This work has been partly financed within the European Horizon 2020 Joint Technology Initiative Shift2Rail through contract no. 730841. The use of AceGen [12] has been very effective in speeding up the implementation of the material models.

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