LOCALIZATION1

The localisation object is used to evaluate the strain and stress concentration tensors \((\boldsymbol{A}_r, \boldsymbol{B}_r)\) as well as the transformation influence tensors \((\boldsymbol{D}_{sr}, \boldsymbol{F}_{sr})\). Several localization rules are available, such as Voigt, Reuss, Mori_Tanaka etc …

Syntax

The syntax depends on the type of localization:

**localization name [ *consistency <value>] \(~\,~\,\) …

*consistency

give the precision, for the TFA consistency checks. In fact, some relations on the TFA method could be checked using the given ratio, which is especially useful for the **localization numeric which allows the user to give his own tensors. It is also useful for debugging. The relations that will be verified are

\[\sum_{r} \boldsymbol{D}_{sr} - \boldsymbol{I} + \boldsymbol{A}_s = 0\]

\[\sum_{r} \boldsymbol{D}_{sr} \boldsymbol{L}^{-1}_r = 0\]

\[\sum_{r} c_r \boldsymbol{D}_{rs} = 0\]

\[c_s \boldsymbol{L}_s \boldsymbol{D}_{sr} - c_r \boldsymbol{D}^T_{rs} \boldsymbol{L}_r = 0\]

Theses relations are verified for each sub-volume \(s\) and the checks are passed when the residual (absolute or relative) is smaller than a preset threshold, which is set in the input-file. The default value of consistency depends on the localization type.

mori_tanaka

This method is used only for 2 sub-volumes, where the matrix is supposed to be the sub-volume 1 and the direction of fibers is 3. The matrix is isotropic.

**localization mori_tanaka \(~\,\) *geometry cylinder <double> <double> | **geometry sphere <double> [ *consistency <double> ] [ *correction stiffness <double> ]

*geometry

geom is sphere or cylinder. r1 and r2 define the geometry of cylinder and sphere.

*consistency

some relations between localisation and influence tensors could be checked using the given ratio.

*correction

a correction for the asymptotic tangent stiffness is done using corrected values for the eigenstrains, instead of using the current tangent stiffness, that needs the evaluation of the instantaneous strain and stress concentration tensors [MS3].

The elastic concentration and influence tensors are evaluated using the Mori-Tanaka method, by the following relations:

\[\boldsymbol{A}_s = \boldsymbol{T}_s \left(\sum_{s=1}^{r} c_r \boldsymbol{T}_r \right)^{-1} \quad {\rm where} \quad \boldsymbol{T}_r = \left[ \boldsymbol{I} + \boldsymbol{S} \boldsymbol{L}_1^{-1} \left( \boldsymbol{L}_r - \boldsymbol{L}_1 \right) \right]^{-1}\]

\[\boldsymbol{D}_{rr} = \left( \boldsymbol{I} - \boldsymbol{A}_r \right) \left( \boldsymbol{L}_r - \boldsymbol{L}_s \right)^{-1} \boldsymbol{L}_r \quad{\rm and}\quad \boldsymbol{D}_{rs} = \left( \boldsymbol{I} - \boldsymbol{A}_r \right) \left( \boldsymbol{L}_s - \boldsymbol{L}_r \right)^{-1} \boldsymbol{L}_s\]

In the matrix, \(\boldsymbol{T}_1 = \boldsymbol{I}\) and \(\boldsymbol{S}\) is the effective Eshelby tensor. The localisation tensor is obtained as follows

\[\boldsymbol{L} = \sum_{r} c_r \boldsymbol{L}_r \boldsymbol{A}_r\]

voigt

This scheme assumes a uniform total strain field among each sub-volume as well as the macroscopic effective material.

**localization  voigt [ *consistency <double>]

Under the dual assumption of \(\boldsymbol{\epsilon}_r = \boldsymbol{E}\), the localisation and influence tensors, using the voigt method, are:

\[\boldsymbol{A}_s = \boldsymbol{I}, \quad \boldsymbol{D}_{rs} = 0 , \quad \boldsymbol{F}_{sr} = \delta_{rs} \boldsymbol{I} - c_r \boldsymbol{L}_s \boldsymbol{L}^{-1} \qquad \forall r, s\]

\[\quad \qquad{\rm and}\quad \boldsymbol{L} = \sum_{r} c_r \boldsymbol{L}_r \boldsymbol{A}_r = \sum_{r} c_r \boldsymbol{L}_r\]

reuss

The Reuss scheme assumes homogeneity of stress among each sub-volume and the macroscopic effective medium.

**localization  reuss [ *consistency <double> ]

The localisation and influence tensors, using the Reuss assumption \(\boldsymbol{\sigma}_r = \boldsymbol{\Sigma}\), are:

\[\boldsymbol{A}_s = \boldsymbol{L}^{-1}_s \boldsymbol{L}, \qquad \boldsymbol{F}_{sr} = 0, \qquad \boldsymbol{D}_{sr} = \boldsymbol{I} - c_r \boldsymbol{A}_s = \boldsymbol{I} - c_r \boldsymbol{L}^{-1}_s \boldsymbol{L} \qquad \forall r, s\]

\[\qquad \qquad \boldsymbol{B}_s = \boldsymbol{I} \qquad{\rm and}\qquad \boldsymbol{L}^{-1} = \sum_{r} c_r \boldsymbol{L}^{-1}_r\]

numeric

The localisation and influence tensors \(A_s\) and \(D_{rs}\) can be directly read from a file, which are given explicitly by the user.

**localization numeric \(~\,\) *number_of_subvolumes <integer> \(~\,\) *A <tensor list> | *localization_tensors_ar_file <file> \(~\,\) *D <tensor list> | *localization_tensors_drs_file <file> [ *consistency <double> ]

The default value of consistency is \(1.e^{-2}\). The tensors A and D are given in a 6x6 representation, with 6 doubles per line and no empty lines between the tensors. The number of sub-volumes should be specified before reading any tensor. Both tensors \(A_r\) and \(D_{rs}\) should be given in the following form:

\[\begin{split}\begin{matrix} D(0,0) & D(0,1) & ... & D(0,n-1) \\ D(1,0) & D(1,1) & ... & D(1,n-1) \\ \vdots & \vdots & \vdots& \vdots \\ D(n-1,0) & D(n-1,1) & ... & D(n-1,n-1) \\ \end{matrix}\end{split}\]

where \(n\) denotes the number of sub-volumes.

polycrystal

This is particularly useful for the approximation of the macroscopic behaviour of the polycrystal for which each grain has a different elastic stiffness or stiffness orientation tensor

**localization polycrystal \(~\,\) C <double>

The elastic concentration and influence tensors are :

\[\boldsymbol{D}_{sr} = (\delta_{sr} - c_r) (\boldsymbol{I} - C \boldsymbol{L}^{-1}) \quad {\rm and} \quad \boldsymbol{E}_{sr} = \boldsymbol{I} + \boldsymbol{D}_{sr} - c_r \boldsymbol{A}_s\]

\[\boldsymbol{L}^{-1} = \sum_{r} c_r \boldsymbol{L}^{-1}_r \quad {\rm and} \quad \boldsymbol{A}_s = \boldsymbol{L}^{-1}_s \boldsymbol{L}\]

tfa_selfconsistent_sphere

The selfconsistent TFA localization is used for N spherical inclusions and anisotropic elasticity. Each sphere can have its own rotation and elasticity.

**localization tfa_selfconsistent_sphere [ *consistency <double>] [ *p_integration <integer>] [ *lhom_max_iterations <integer> ]

*p_integration

The number of points for the numerical integration on the surface of the unit sphere, the default value is 25.

*lhom_max_iterations

Maximum number of iterations to find the homogeneised medium, where the default value is 10.

*consistency

The default value is \(1.0e^{-10}\).

Using the selfconsistent method, the elastic concentration and influence tensors are given by the following relations:

\[\boldsymbol{D}_{sr} = (\boldsymbol{I} - \boldsymbol{A}_s) ( \boldsymbol{L}_s - \boldsymbol{L} )^{-1} (\delta_{sr} \boldsymbol{I} - c_r \boldsymbol{A}_r^T ) \boldsymbol{L}_r ,\quad \boldsymbol{A}_r = (\boldsymbol{L^*} + \boldsymbol{L}_s )^{-1} (\boldsymbol{L^*} + \boldsymbol{L})\]

\[\boldsymbol{F}_{sr} = (\boldsymbol{I} - \boldsymbol{B}_s) ( \boldsymbol{L}^{-1}_s - \boldsymbol{L}^{-1} )^{-1} (\delta_{sr} \boldsymbol{I} - c_r \boldsymbol{B}_r^T ) \boldsymbol{L}_s^{-1} ,\quad \boldsymbol{L} = \left( \sum_{r} c_r (\boldsymbol{L^*} + \boldsymbol{L}_s )^{-1} \right) - \boldsymbol{L^*}\]

where \(\boldsymbol{L^*} = \boldsymbol{P} - \sum_{r} c_r \boldsymbol{L_s}\) is the Hill’s constaint tensor and \(\boldsymbol{P}\) is the Hill polarization tensor [MS4, MS5, MS6] .

The steps to evaluate the localization tensor \(\boldsymbol{L}\) are:

• First estimate : \(L^0 = \sum_{r} c_r \boldsymbol{L_s}\)

• Estimate \(\boldsymbol{L}(\boldsymbol{P})\)

• Calculate \((max |Ln-Ln+1| < consistency)\)

• Main loop until the \(|Pn-Pn+1| < consistency\) or until nmax is reached.

multilayer

Multilayer localization only valid for 3D problems and tensor size 6. The geometry of the layer is given by : \((1,2)\) plane of the layers and \(3\) direction perpendicular to the layers.

**localization  mutlilayer [ *consistency <double> ]

Numerical calculations of localisation and influence tensor

It is possible to calculate the influence and localisation tensors using a numerical method. Numerical approximation of localization and eigenstrain influence tensors in the literature are given by Dvorak and Teply [MS7] as well as Paley and Aboudi [MS8]. The theoretical developments are not presented here.

Usage : Zrun -micmacname.inp.

These tensors can be read in Multimat using **localization numeric

Syntax

****micmac ***no_sqrt2 ***local local ***compute_A ***compute_B ***grad options ***compute_D ***check_consistency

***no_sqrt2

do keep the sqrt2 in the non diagonal terms. Please use this option.

***local

specify the name of the localisation problem (see FE\(^2\)) method

***compute_A

to compute the strain localisation tensor.

***compute_B

to compute the stress localisation tensor.

***grad

specify the strain apply to the local problem to calculate (see example).

***compute_D

to compute the strain influence tensors.

***check_consistency

to verify the relations between A and D.

Example

A 2D generalized plane strain example.

****micmac
  ***no_sqrt2
  ***local local
  ***compute_A
  ***compute_B
  ***grad 4 1. 0. 0. 0.
  ***grad 4 0. 1. 0. 0.
  ***grad 4 0. 0. 1. 0.
  ***grad 4 0. 0. 0. 1.
% if 3D problem, use :
% ***grad 6 1. 0. 0. 0. 0. 0.
% ***grad 6 0. 1. 0. 0. 0. 0.
% ***grad 6 0. 0. 1. 0. 0. 0.
% ***grad 6 0. 0. 0. 1. 0. 0.
% ***grad 6 0. 0. 0. 0. 1. 0.
% ***grad 6 0. 0. 0. 0. 0. 1.
  ***compute_D
  ***check_consistency
****return