Polycrystalline plasticity approach

This section describes procedures used to determine the full constraint of the Taylor model, and its relaxed version.

6.6.1 Materials preparation

The polycrystalline plasticity models consider texture as the main source of anisotropy. Therefore, the initial texture must be measured in the form of an ODF, by conducting a texture goniometer measurement. However, before proceeding with the measurement, such material must be prepared.

Through-thickness texture gradient is assumed to be negligible [118,143,155,191] – i.e. the texture in the mid-thickness layer is the same as that in the outer surface of the sample. However, to ensure a representative texture, the sample must be etched to its centreline layer. Deriving the texture at the mid-thickness layer has the advantage of avoiding those layers where processing rolls contact the specimen; thus, potential surface heterogeneities are eliminated [192].

To measure the material’s texture at the mid-thickness layer, a square sample of 10×10 mm is obtained. The surface of the sample needs to be grinded and polished using grinding papers (700, 1200, 4000 [~7µm]), diamond pastes (3 µm and 1 µm), and a final stage with an oxide polishing system with SiO2 (0.25 µm), to avoid any deformed material.

6.6.2 X-ray (Texture, pole figures, ODF)

6.6.2.1 X-ray experiment

The X-ray technique is used to measure macro-texture with the so-called Schultz reflection method [151]. This method allows a pole figure to be obtained. However, some useful texture information may be lost, and to overcome this uncertainty an ODF is used [147]. Using an X-ray pole figure goniometer is more rapid than other techniques and easily automated, as well as being inexpensive in acquisition and maintenance [145].

Texture or preferred orientation of crystallites is an intrinsic aspect of metals, and the physical properties of the materials, such as strength, toughness, etc., will be affected by the texture, particularly in the anisotropy of these properties [145].

One conventional technique used to measure macro-texture – averaged orientation data from many grains – of the material is the X-ray technique, which is much easier in terms

of access than neutron sources [151]. Also, it is a reliable and accurate measurement for measuring Taylor models, as many studies have demonstrated – e.g. [7,61,125,141,193].

The number of grains sampled by the X-ray technique is larger than those able to be sampled using the EBSD method, which is based on an electron source [194]. However, it has been established that the EBSD is also an alternative for the X-ray technique when deriving a macro-texture result [194]. Thus, for the purpose of the thesis, the ability to sample more grains would characterise the sheet more efficiently. This measurement can also be made via neutron diffraction, which is an alternative means for measuring macro- texture of the materials. However, the X-ray machine will be able to produce a reliable result for the purposes of this thesis.

Measurements were conducted on a Siemens D500 X-ray goniometer at the Department of Materials Engineering, University of Leuven. The goniometer is equipped with a copper anode – with a wavelength of 1.54051 Ao – operating at 40 kV and 40 mA.

Under such operating conditions, for AA6111-T4 and AC600, four incomplete pole figures {111}, {200}, {220} and {311} were measured, while for the BCC metals DX54D and H220BD+Z, pole figures {110}, {200}, {211} and {103} were measured.

Powder samples of 99% pure aluminium and 99% pure iron with particle sizes of 10 microns must be measured for background and defocusing correction procedures. The small sizes were chosen in order to ensure the absence of texture – i.e. random texture.

6.6.2.2 Corrections

Defocusing and background correction procedures for the raw data or pole figures measured by the D500 X-ray goniometer must be performed using MTM-FHM software developed by Van Houtte [154]. This is used to process incomplete pole figures measured with the X-ray diffraction technique using the back reflection method employing a texture goniometer, in order to produce an ODF using Euler angles. This system can be used for materials with cubic crystal structure and orthorhombic sample symmetry, as is the case for cold-rolled aluminium and steel sheets. Another feature of this software is its ability to calculate yield loci derived from the full-constraint Taylor model and its relaxed version, both of which were described in Chapter 4.

The background and defocusing are due to incoherent scattering and fluorescence in the sample, and increasing sample tilt, respectively [151]. Another error intrinsic to the experiment is that computed ODFs contain truncation and ghost errors that require

correction [146,151]. There is a need to truncate the series expansion of pole figure as well as the ODF, which leads to a broadening of texture peaks and a minimising of some intensities near strong texture components. Missing or wrong intensities that appear are termed negative or positive ghosts, respectively [151]. Such errors must be corrected by the MTM-FHM software.

6.6.2.3 ODF

After obtaining corrected measurements of the pole figures, the texture is presented by an ODF. For FCC metals, the ODF is measured from four pole figures {111}, {200}, {220} and {311}, while for BCC metals, it is measured from pole figures {110}, {200}, {211} and {103}.

To retrieve an ODF from the pole figure, various methods can be used [144–146,151], such as the WIMV method developed by Williams, Imhof, Matthies, and Vinel, the vector method, and the maximum entropy method, which function in direct space and use tomography algorithms. In contrast, other methods such as the most well-established technique – namely, the harmonic method introduced by Bunge (1965) – work in Fourier space. The MTM-FHM calculates the ODF based on the harmonic method. It is important to note that all of these methods yield similar results [145]. Although the direct methods implicitly address some of the problems associated with measurement, such as ghost errors, the harmonic method is more rapid and is considered reliable [145,151]. Also, some texture-related properties – e.g. plastic anisotropy – can be calculated using C- coefficients. Moreover, using the harmonic method yields an easier normalisation procedure of the pole figures [151].

6.6.2.4 Yield loci

Based on measured texture data (ODF), yield loci of the materials will be calculated using the polycrystalline plasticity models: the Taylor full-constraint model and the relaxed pancake model. These models are implemented in the MTM-FHM software. Both Taylor models are required to compute the CTFP model, whilst the full-constraint Taylor is the only model used for the development of the new model, referred to as the CTF.

From the Taylor models, the CTFP is computed. Then, the CTFP must be validated for different aluminium grades. This process is easily carried out by using data points of the combined model (CTFP), with the assistance of a uniaxial tensile test directly in the BBC2005 model, and a comparison of this with the one measured via the experimental work – involving all three basic mechanical tests: tensile, bulge, and compression.

6.6.3 Comparison between the two approaches

Another objective of this thesis is to carry out a comparison between the two approaches – the phenomenological approach (BBC2005 – von Mises – Hill ‘48) and the polycrystalline plasticity approach (FC, RC, CTFP, and CTF) – in terms of accuracy. This comparison is outlined in Chapter 7.