General_TFA_Method#

In this section, we recall the main line of the Transformation Field Analysis method introduced by Dvorak et al. [Dvorak, 1992]. This method is very similar to the FE\(^2\) one (1. localization, 2.constitutive equations at the local scale, 3. homogenization), but it is technically very different. The TFA method is based on concentration and influence tensors and considers the plastic strain and thermal expansion as eigenstrains of the same nature. The composite scale is divided into a number of sub-volumes (N) inside which the mechanical fields are assumed to be uniform. A sub-volume \(r\) is thus a homogeneous part of the representative volume element. The TFA models link the local fields (\(\boldsymbol{\sigma}_r\) and \(\boldsymbol{\epsilon}_r\)) to the macroscopic ones (\(\boldsymbol{\Sigma}\) and \(\boldsymbol{E}\)) with the following relations :

(573)#\[ \begin{align}\begin{aligned}:label:dvor2s\\\boldsymbol{\sigma}_r=\boldsymbol{B}_r:\boldsymbol{\Sigma}-\sum_{s=1}^{N}\boldsymbol{F}_{{rs}}:\boldsymbol{L}_s: \boldsymbol{\gamma}_s\end{aligned}\end{align} \]
(574)#\[ \begin{align}\begin{aligned}:label:dvor1eb\\\boldsymbol{\epsilon}_r=\boldsymbol{A}_r:\boldsymbol{E}+\sum_{s=1}^{N}\boldsymbol{D}_{rs}: \boldsymbol{\gamma}_s\end{aligned}\end{align} \]

with

(575)#\[\boldsymbol{\gamma}_r = \boldsymbol{\epsilon}_r - \boldsymbol{\epsilon}^e_r = \boldsymbol{\epsilon}_r^p + \epsilon_r^{th}\]

where \(\boldsymbol{\gamma}_r\) is the formal eigenstrain. \(\boldsymbol{A}_r\) and \(\boldsymbol{B}_r\) are the elastic strain and stress concentration tensors respectively (fourth rank tensors) of the sub-volume \(V_r\). \(\boldsymbol{F}_{{rs}}\) and \(\boldsymbol{D}_{rs}\) are the transformation influence tensors (fourth rank tensors); the subscript \(rs\) denotes the influence of the sub-volume \(s\) on the sub-volume \(r\). These are determined by solving a set of linear problems (6 for the concentration tensors and 6*N for the influence tensors) by a finite element method. These tensors are function of the elastic behavior of each sub-volume and the shape of the VER. \(\epsilon_s^{th}\) and \(\epsilon_{s}^p\) denote the uniform thermal and plastic strain of the sub-volume s.

In the general case with varying local elastic behavior (temperature or damage induced), the formal eigenstrain \(\boldsymbol{\gamma}_r\) is determined from the specification of the current state of the sub-volume, which can be generalized to take into account the thermal and damage effects on the elastic behavior. This generalized eigenstrain and the local behavior are obtained as follows,

(576)#\[\boldsymbol{\gamma}_r = \boldsymbol{\epsilon}_r - \boldsymbol{\epsilon}^e_r = \boldsymbol{\epsilon}_r^p + \boldsymbol{\epsilon}_r^{th} + \boldsymbol{\epsilon}_r^d \qquad\quad{\rm and}\quad\qquad \boldsymbol{\sigma}_r = \boldsymbol{\widetilde{L}(T)}_r : (\boldsymbol{\epsilon}^e_r) = \boldsymbol{L}^0_r: (\boldsymbol{\epsilon}_r - \boldsymbol{\gamma}_r )\]

where \(\boldsymbol{\widetilde{L}(T)}_r\) is the thermally dependent and/or damaged elastic stiffness, which is related to the actual state of the sub-volume \(r\) and \(\boldsymbol{L}_s^0\) is the initial undamaged matrix, which is independent of temperature.

(577)#\[\begin{split}\left\{ \begin{array}{lc} \boldsymbol{\gamma}_r = \boldsymbol{\epsilon}_r^p + \boldsymbol{\epsilon}_r^{th} \qquad & \qquad {\rm if}\quad \boldsymbol{\widetilde{L}(T)}_r = \boldsymbol{L}_r\\ \boldsymbol{\gamma}_r = \boldsymbol{\epsilon}_r^p + \boldsymbol{\epsilon}_r^{th} + \boldsymbol{\epsilon}_r^d \qquad & \qquad {\rm if}\quad \boldsymbol{\widetilde{L}(T)}_r \neq \boldsymbol{L}_r \end{array} \right.\end{split}\]

and the eigenstrain related to the damage effect can be rewritten as follows :

(578)#\[\boldsymbol{\epsilon}_r^d=(\boldsymbol{I}-\boldsymbol{\widetilde{L}(T)}_r^{-1}:\boldsymbol{L}_r^0):\epsilon_e = \boldsymbol{\epsilon}_r^g=(\boldsymbol{\widetilde{S}}_r-\boldsymbol{S}^0_r):\boldsymbol{\sigma}_r\]

where \(\tilde{\boldsymbol{S}}_r\) and \(\boldsymbol{S}^0_r\) are respectively the actual and initial compliance of the sub-volume r.

It is important to note that damage or thermal change in the elastic modulus makes the eigenstrain not only a function of the plastic and thermal strain but also a function of the elastic strain. This dependence along with rate independent plasticity will make a two way coupling between all the local total strain rates and all the eigenstrains. In addition, various physical effects must be take into account, like the interaction between the interface on dislocation movement and the effects of the incoherent interface with the discontinuity of the displacements, in order to obtain a good local strain/stress redistributions, as well as a correct stress-strain behavior.

The TFA method is then considered in conjunction with the “Generalized Eigenstrain Model”, where corrected constitutive eigenstrain \(\beta_r\) is introduced and expressed in terms of the formal eigenstrain plus a correction:

(579)#\[\boldsymbol{\beta}_r = \boldsymbol{\gamma}_r + \boldsymbol{\xi}_r\]

Here the eigenstrain \(\beta_r\) is to be considered as an internal variable which must be assigned an evolution law, which can be tuned to fit the available data.

This evolution law is associated either with a stress localization or with a strain localization law but not both. In this case, the stress localization and the strain localization can be written as follows:

Strain localization :

(580)#\[\begin{split}\left\{ \begin{array}{lcl} \displaystyle{ \boldsymbol{\epsilon}_r} & = & \displaystyle{ \boldsymbol{A}_r:\boldsymbol{E}+\sum_{s=1}^{n} \left[ \boldsymbol{D}_{sr}: \boldsymbol{\gamma}_s + \boldsymbol{E}_{sr}: \boldsymbol{\xi}_s \right] \quad {\rm where} \quad \boldsymbol{E}_{sr} = \boldsymbol{D}_{sr}}\\\\ \displaystyle{ \boldsymbol{\sigma}_r } & = & \displaystyle{ \boldsymbol{B}_r:\boldsymbol{\Sigma} - \sum_{s=1}^{n} \boldsymbol{F}_{sr}: \boldsymbol{L}_s:\boldsymbol{\beta}_s - \boldsymbol{L}_r:(\boldsymbol{\gamma}_r - \boldsymbol{\beta}_r) - \boldsymbol{B}_r : \sum_{s=1}^{n} c_s \boldsymbol{L}_s:(\boldsymbol{\gamma}_r - \boldsymbol{\beta}_r)} \end{array} \right.\end{split}\]

Stress Localization :

(581)#\[\begin{split}\left\{ \begin{array}{lcl} \displaystyle{ \boldsymbol{\epsilon}_r} & = & \displaystyle{ \boldsymbol{A}_r:\boldsymbol{E}+\sum_{s=1}^{n} \left[ \boldsymbol{D}_{sr}: \boldsymbol{\gamma}_s + \boldsymbol{E}_{sr}: \boldsymbol{\xi}_s \right] \quad {\rm where} \quad \boldsymbol{E}_{sr} = \boldsymbol{D}_{sr} - \delta_{sr} \boldsymbol{I} - c_r \boldsymbol{A}_s} \\\\ \displaystyle{ \boldsymbol{\sigma}_r} & = & \displaystyle{ \boldsymbol{B}_r:\boldsymbol{\Sigma} - \sum_{s=1}^{n} \boldsymbol{F}_{sr}: \boldsymbol{L}_s:\boldsymbol{\beta}_s} \end{array} \right.\end{split}\]

where \(c_r\) is the volume fraction of the sub-volume r (\(c_r=V_r/V\)).

The macroscopic behavior (macroscopic strain \(\boldsymbol{\Sigma}\) and stress \(\boldsymbol{E}\)) can be obtained by averaging the local stresses (\(\boldsymbol{\sigma}_r\)) and strain (\(\boldsymbol{\epsilon}_r\)) on the RVE :

(582)#\[\boldsymbol{\Sigma}=\biggl<\boldsymbol{\sigma}(x)\biggl>_V=\sum_rc_r\boldsymbol{\sigma}_r \qquad,\qquad \boldsymbol{E}=\biggl<\boldsymbol{\epsilon}(x)\biggl>_V=\sum_rc_r\boldsymbol{\epsilon}_r\]

Syntax#

***behavior general_tfa \(~\,\) **material  volume_fraction name | **material_in_file <file> *file file_name \(~\,\) [ *integration method ] \(~\,\) [ *rotation <ROTATION> ] | [ *rotation_list <file> ] \(~\,\) [ *volume_fraction_file <file> ] **material etc … \(~\,\) **localization <LOCALIZATION1> [ **solver tfa_implicit1 ] [ **reference_temperature <double> ] [ **eigenstrain <EIGENSTRAIN> ] ***return

**material

specify the different sub-volumes and its volume fractions. The number of **material is equal to the number of sub-volumes.

**material_in_file

give the name of the external file where the different sub-volumes are defined. This file name is relative to the current working directory. Using **material, just one sub-volume can be specified whereas when **material_in_file is used, many sub-voumes can be defined.

**localization

specify the localization method for determining the strain and stress concentration tensors \((\boldsymbol{A}_r, \boldsymbol{B}_r)\) as well as the transformation influence tensors \((\boldsymbol{D}_{sr}, \boldsymbol{F}_{sr})\)

**eigenstrain

define the evolution law for the eigenstrain variable, which is considered as an internal variable. It has the dimension of a strain.

**solver

A consistent tangent matrix is provided with this model if implicit integration is chosen, and the sub-materials have a consistent tangent.

The options *rotation_list or **volume_fraction_file only available when using **material_in_file.

Example#

***behavior general_tfa
  **solver tfa_implicit1
  **localization polycrystal
     C 50000.0
  **eigenstrain beta
     D 10000.0
  **material  0.2 grain1
   *file poly_trac_2.inp 2
   *rotation -149.676 15.61819 154.676
   *integration theta_method_a 1. 1.e-9 200
  **material 0.3 grain2
   *file poly_trac_2.inp 2
   *rotation -210.324 15.61819 205.324
   *integration theta_method_a 1. 1.e-9 200
  **material 0.5 grain3
   *file poly_trac_2.inp 2
   *rotation -94.2711 35.46958 171.271
   *integration theta_method_a 1. 1.e-9 200
***return

This is an example of the TFA behavior with three sub_volumes (grain1, grain2 and grain2) with respective volume fractions 0.2, 0.3 and 0.5, using the polycrystal localization rule.