High Temperature Deformation of Coarse Grain

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Diffraction Peak Profile Analysis DPPA for studying Plastic Deformation T.H. Simma, P.J. Withersb, J Quinta da Fonsecab a) Institute of Structural Materials, College of Engineering, Swansea University. b) Materials Science Centre, University of Manchester

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Outline

1. What is Diffraction Peak Profile Analysis?

2. What are the parameters we obtain from DPPA • How useable are they for understanding plastic deformation?

3. How DPPA can be used with plasticity models • Can this be a tool to verify models

• that predict slip system activity? • And incorporate work-hardening? enter RR clearance number or meeting details enter RR clearance number or meeting details

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Diffraction Peak Profile Analysis DPPA

Detector

A perfect crystal would cause no broadening,

Compression

Strain Broadening: Variations in d-spacing broaden the peak, e.g. around dislocations

Tension

Size Broadening: When the crystal size is less than about < 0.5 μm there is not enough interference to create a delta peak,

g diffraction vector

θ Source (of wavelength λ)

d

Braggs Law

2d.sinθ = λ


DPPA Methods Available DPPA quantifies defects to perfect crystal structure, mainly by dislocations Methods 1. Full-width 2. Williamson-Hall: Full-width (β), crystal size (V), position of peak (g=1/d), microstrain (ε) 3. Fourier Methods: (e.g. Warren-Averbach)

ln( AL )   ln( ALS )  2 2 g 2 L2  L2

Fourier Coefficients (AL), size coefficients (AS), Fourier length (L) 4. Others: Variance method, strain-field method etc

Outputs: 1. Dislocation density 2. Dislocation arrangement 3. Dislocation slip systems present 4. Crystal size 5. Others e.g. planar fault density enter RR clearance number or meeting details enter RR clearance number or meeting details

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The Experiment

Alloys FCC a) Austenitic stainless steel (304 and 316) b) Nickel 200 (commercially pure >98% Ni) HCP a) Ti-6Al-4V b) Ti-CP grade 2 (>99% Ti) Lab. X-ray (Nickel, Stainless Steel)

Synchrotron X-ray (Titanium)


Change in full-width

Modified Williamson-Hall 𝐾𝑆𝑐 ҧ ҧ 𝛽 𝑔 = + 𝑓𝑀 𝑔 𝜌𝐶ℎ𝑘𝑙 + 𝑂 𝑔2 𝐶ℎ𝑘𝑙 𝐷 Taylor equation σ = σ0 + α′Gb ρ Therefore, Full-width should be proportional to flow stress

- Full-width (β), crystal size (D), position of peak (g=1/d), dislocation density (ρ), contrast factor (C), O(…), fM associated with dislocation arrangement - σ is the flow stress, σ0 the friction (yield) stress, α’ a constant, G the shear modulus, b the magnitude of the Burgers vector enter RR clearance number or meeting details enter RR clearance number or meeting details

Dislocation densities

σ = σ0 + α′Gb ρ

Alternative given by van Berkum 1994

Williamson-Hall

Fourier Methods

• The simpler Williamson-Hall methods better predict changes in dislocation density • The Fourier methods encounter problems at low applied stresses

Expected


Size in DPPA Stainless steel

Nickel Changes in size of other alloys not expected. • Alternative is only method to differentiate

Of the alloys, nickel is the only one that should develop a size structure that can be measured by DPPA

Conclusion: • Size is an artefact or representative of dislocations breaking coherency • Makes parameter difficult to use

The ratio of sizes of nickel with reported TEM to DPPA data: • Around 1:2 (for mWH-3) to 1:8 (for Fourier), with an average ratio of 1:7 • Consistent with other measurements, showing sizes by DPPA are smaller enter RR clearance number or meeting details enter RR clearance number or meeting details

Dislocation Arrangement (dipole character M) Nickel ε = 0.1 [1]

SS ε = 0.15 [2]

Dislocation arrangement, or the arrangement of dislocations e.g. into tangles, cells etc, is given by one parameter the dipole character or M Dislocations arranged in cells screen the strain field of over dislocations causing less broadening • Are the changes real or an artefact and source of errors? • Changes consistent with what other find- but why are Ni and SS so similar? • In WH methods (better correlation for dislocation density) we assumed M is constant • Again parameter difficult to use

[1] C. Keller, Mech. Mater. 42 (2010) 44–54. [2] T.S. Byun, J. Nucl. Mater. 321 (2003) 29–39.


Slip System Activity (Contrast factor C) Cubic

Hexagonal close packed (HCP)

Screw

Edge

• Transition from edge to screw & more • Dominant expected edge for Nickel • But problems because of the simplicity • Both consistent with work-hardening of the approach models Contrast factor / slip system predictions are potentially useful but care needed enter RR clearance number or meeting details enter RR clearance number or meeting details

3

Experiment 2- Polycrystal Plasticity Models - Maybe we are doing too much averaging? - So instead consider different texture components separately

High Resolution Powder Diffractometer (HRPD), ISIS, STFC, Oxfordshire, UK enter RR clearance number or meeting details enter RR clearance number or meeting details

Slip systems present (the contrast factor)

𝛽 𝑔 =

𝐾𝑆𝑐 + 𝑓𝑀 𝑔 𝐷

ҧ ҧ 𝜌𝐶ℎ𝑘𝑙 + 𝑂 𝑔2 𝐶ℎ𝑘𝑙

Full-width (β), crystal size (D), position of peak (g=1/d), dislocation density (ρ), contrast factor (C), O(…) and fM associated with dislocation arrangement

Contrast factor -> 0 g

Borbely et al 2003 J Appl Cryst

Contrast factor -> max

g

Contrast factor depends on the relative direction of the Burgers vector (b), the line vector and the line vector relative to the diffraction vector (g)

b


Taylor Model Predictions


Broadening Anisotropy Stainless Steel - Fatigue

Stainless Steel - Tensile

Nickel - Tensile

Measured

Full-width / g • Correlation between measured & predicted •

Predicted But something else going on


Taylor factor and dislocation arrangement

Dislocation arrangement is orientation dependent and related to Taylor factor enter RR clearance number or meeting details enter RR clearance number or meeting details

PV Mixing parameter

Peak tails and arrangement

PV = how close peak is to Lorentzian (=1) or Gaussian (=0)

Angle between tensile direction and g • Correlation between Taylor factor and shape for Nickel but not Steel • Suggests arrangement details of dislocations can be obtained from DPPA enter RR clearance number or meeting details enter RR clearance number or meeting details

Changing other parameters

(A) Plasticity approach with varying dislocation density in grains (B) Homogeneous approach with varying dislocation density in grains (C) Plasticity approach with the same dislocation density in all grains (D) Homogeneous approach with the same dislocation density in all grains

Lots of possible ways full-width can change with angle. Lets say: FW = ∑ki x FWi And find ki by fitting to the data enter RR clearance number or meeting details enter RR clearance number or meeting details

Extra Parameters Strain

Edge

Dislocation Mobile

Percentage density

Dislocation

% (k3)

% (k2)

variation % (k1)

Stainless

Fatigue

Steel

*

Nickel

53%

33%

100%

10%

49%

0%

33%

16%

56%

0%

50%

30%

52%

0%

24%

10%

100%

33%

23%

30%

94%

24%

15%

SS

Nickel

More edge for Niexpected Ni has broadening related to Taylor factor to account for arrangement changes Ni has less mobile dislocation e.g. from cross-slip (e.g. a flatter change of FW) enter RR clearance number or meeting details enter RR clearance number or meeting details

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Conclusions Parameters from DPPA for deformed metals: 1. Dislocation density •

Can get values expected- but results are methodology and experimentally dependent

2. Dislocation arrangement and crystal size •

In many cases of limited practical use,

•

Instead need to consider selected orientations

3. Slip systems •

DPPA can give information of slip systems and verify plasticity models

•

But other causes of broadening means the task is not straightforward Diffraction Peak Profile Analysis (DPPA) is a useful technique • But care must be taken • And existing formulations extended

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