the Lankford coefficients, the R-values, of a sheet from its orientations or its ODF

Syntax

R, M, minM = calcLankford(ori, sS)

R at 0, 5, …, 90 degree to the rolling direction.

R, M, minM = calcLankford(ori, sS, theta, weights=w)

Weighted orientations, chosen angles.

R, M, minM = calcLankford(odf, sS, rho=np.linspace(0, 1, 21))

From an ODF, a finer ratio.

Description

The R-value is the ratio of the true width strain to the true thickness strain of a strip pulled at the angle theta to the rolling direction. For every theta the strip is strained along the pull and contracts by the fraction rho across it and 1 - rho through the thickness; the Taylor model gives the normalised plastic work M of every rho, averaged over the texture, and the strip takes the rho of least work, R = rho / (1 - rho).

The mean Rbar of R over the directions tells how well a sheet resists thinning, Rbar >= 1 favouring deep drawing; the planar anisotropy deltaR indicates the fourfold earing of a cup.

The angles theta may also be given as a positional array after sS, as MATLAB.

Input

`x`	orientation, the crystal orientations of the sheet, or SO3Fun, its ODF
`sS`	slipSystem, or a TaylorModel, the systems the crystals deform by

Keywords

keyword	default	description
`theta`	`None`	the angles of the pull to the rolling direction in radians
`rho`	`None`	the fractions of the contraction across the strip, between 0 and 1
`RD`	`None`	vector3d, the rolling direction, the pull at theta = 0
`ND`	`None`	vector3d, the normal of the sheet, orthogonal to RD
`weights`	`None`	the weights of the orientations, the area of grains for instance
`bandwidth`	`16`	the harmonic bandwidth the Taylor factor and the ODF are expanded to (an ODF)

Output

`R`	the Lankford coefficient at the least plastic work, one per theta
`M`	the normalised plastic work, an array of rho by theta
`minM`	the least normalised plastic work, one per theta

Used on

Lankford

MATLAB’s calcLankford

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/calcLankford_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.

calcLankford