Estimation of the Full Nye Tensor by EBSD-Based Dislocation Microscopy Thomas Hardin1,a, Brent L. Adams1,b, David T. Fullwood1,c, and Robert H. Wagoner2,d 1
Brigham Young University Department of Mechanical Engineering Provo, Utah 84602, USA 2 The Ohio State University Department of Materials Science and Engineering Columbus, Ohio 43210, USA a
[email protected], b
[email protected], c
[email protected], d
[email protected] Keywords: EBSD, HR-EBSD, Nye Tensor, Stereology. Abstract. An extension to a previously published, novel stereological method is reported which infers experimentally inaccessible components of the Nye GND tensor. Limitations imposed by electron-opacity of metals prevent direct measurement of four components of the Nye tensor, but it is possible to use additional experimentally-obtainable information in connection with underlying field equilibrium equations to estimate these additional components. This approach uses derivatives to the infinitesimal elastic distortion tensor to reduce error imposed by pattern center inaccuracy. Introduction The distribution of dislocations can be quantified on the continuum level by Nye’s tensor αij. Nye’s tensor can be calculated for a finite volume of material by integrating along all dislocations’ lines in the volume: 𝛼𝑖𝑗 ≡
1 𝑉
𝐿
𝑏𝑖 𝑡𝑗 d𝑠
(1)
where V is the element’s volume, b is a dislocation’s Burger’s vector, and t is the unit vector tangent to the dislocation line [1]. So the first subscript of αij represents the direction of the Burger’s vector and the second subscript represents the dislocation line direction. It is possible to directly recover three components of Nye’s tensor by HR-EBSD microscopy; it is also possible to estimate two additional components [2]. This leaves four remaining components that are experimentally inaccessible by standard HR-EBSD methods. In an article in ICHMM 2011 conference proceedings, Hardin et al. reported a method for estimating the four experimentally inaccessible components of Nye’s tensor [3]. This method used the relationship between dislocation density and the measured infinitesimal elastic strain field in a lattice to generate a system of linear equations that could then be solved for the unknown Nye’s tensor components. One shortcoming of that work was that absolute elastic strain measurements are sensitive to small inaccuracies in locating the EBSD pattern center (this is described in detail in [4]). This paper extends the work of Hardin et al. to eliminate this pattern-center sensitivity, and also includes additional experimentally-accessible information to further improve the results that can be obtained. Method The method is similar to that reported in [3]; the difference is that it works exclusively with derivatives of the infinitesimal elastic distortion tensor instead of infinitesimal elastic strain and Nye dislocation density. This is beneficial since derivatives of elastic distortion are robust to pattern center inaccuracies that degrade distortion measurements [4]. Also, by treating the full distortion
tensor instead of elastic strain (the symmetric part of the distortion tensor), information is retained from the antisymmetrical part that was not considered in [3]. Fundamental Equations. The infinitesimal elastic distortion tensor β is defined in terms of partial derivatives of the elastic displacement vector u of a lattice: 𝛽𝑖𝑗 =
𝜕𝑢𝑖 = 𝑢𝑖,𝑗 𝜕𝑥𝑗
(2)
where a subscript after a comma denotes a partial derivative. The fundamental equation of this method is based on an expression derived in [5], and relates the infinitesimal elastic distortion at a given point, x, in the lattice to the dislocation density and infinitesimal elastic distortion at all other points, x′, in the lattice: 𝛽𝑘𝑙 𝐱 =
V
𝑟 ∈𝑙𝑛 ℎ G𝑘𝑖 ,𝑗 𝐱 − 𝐱 ′ 𝐶𝑖𝑗𝑚𝑛 𝛼𝑚ℎ 𝐱 ′ 𝑑𝐱 ′
(3) ′
− V
G𝑘𝑖 ,𝑙𝑗 𝐱 − 𝐱 𝐶
′
𝑖𝑗𝑚𝑛
′
′
′
𝐱 𝛽𝑚𝑛 𝐱 𝑑𝐱 + 𝜁𝑘𝑙
where G is a Green’s function, as described by Mura [6], and ϵ is the permutation operator. Repeated subscripts on one side of the equation denote summation (per the Einstein subscript convention). The term ζ is a placeholder for a singularity term related to boundary conditions, and is not treated here. Teodosiu reveals the relationship between the Nye dislocation density and the infinitesimal elastic distortion at a lattice point [7]: 𝛼 = 𝑐𝑢𝑟𝑙 𝛽 → 𝛼𝑙𝑗 =∈𝑛𝑚𝑗 𝛽𝑙𝑛 ,𝑚
(4)
which indicates that dislocation density can be expressed completely in terms of partial derivatives of elastic distortion. The corollary of this is that if all the partial derivatives of the elastic distortion tensor are known, the full Nye’s tensor may be recovered. Since EBSD is fundamentally a surface characterization method, it is not possible to measure terms of βkl,h where h=3. If these terms were known, the full Nye’s tensor could be constructed by application of eqn. (4). Equation (3) is simplified by considering an isotropic approximation, where C′ (the local perturbation in elastic stiffness) goes to zero. In an isotropic case with no external loading, the singularity term ζ is also presumed to be zero. Substituting equation (4) into the isotropic version of equation (3), applying the properties of the permutation operator, and taking a partial derivative with respect to xh yields equation (5): 𝛽𝑘𝑙 ,ℎ 𝐱 =
V
𝑟 ∈𝑙𝑛 ℎ G𝑘𝑖 ,𝑗 ℎ 𝐱 − 𝐱 ′ 𝐶𝑖𝑗𝑚𝑛 𝛽𝑚𝑛 ,𝑙 𝐱 ′ − 𝛽𝑚𝑙 ,𝑛 𝐱′
𝑑𝐱 ′
(5)
which can be discretized to suit a grid of EBSD points: ∞ 𝑥𝑦𝑧 𝛽𝑘𝑙 ,ℎ
≈𝑑
3
∞
∞
𝑟 ∈𝑙𝑛 ℎ 𝐶𝑖𝑗𝑚𝑛
𝑥−𝑥 ′ ,𝑦−𝑦 ′ ,𝑧−𝑧 ′
G𝑘𝑖 ,𝑗 ℎ
𝑥′ 𝑦 ′ 𝑧′
𝛽𝑚𝑛 ,𝑙
𝑥′ 𝑦 ′ 𝑧′
− 𝛽𝑚𝑙 ,𝑛
(6)
𝑥 ′ =−∞ 𝑦 ′ =−∞ 𝑧 ′ =−∞
where d is the grid spacing and superscripts represent data point locations. Since this is a volume integral, it is necessary to make some assumption of columnarity to evaluate this expression. One possible option would be to assume that dislocation density does not vary in the z-direction.
Experimental Considerations. As described by Wilkinson [8], while the off-diagonal components of β can be measured absolutely without applying a traction-free boundary condition, the diagonal (normal) components can only be measured relative to each other. That is, HR-EBSD methods enable us to measure two values (Δ1 and Δ2) in addition to the off-diagonal (shear and rotational) components of β at each point. These are defined in eqns. (7,8): ∆1 𝐱 = 𝛽11 𝐱 − 𝛽22 𝐱
(7)
∆2 𝐱 = 𝛽22 𝐱 − 𝛽33 𝐱 .
(8)
Taking partial derivatives of eqns. (7,8) yields the following equations: ∆1,ℎ 𝐱 = 𝛽11,ℎ 𝐱 − 𝛽22,ℎ 𝐱
(9)
∆2,ℎ 𝐱 = 𝛽22,ℎ 𝐱 − 𝛽33,ℎ 𝐱
(10)
where h=1 or h=2. This provides four additional equations per point in terms of derivatives of β. Application of Equation (6). Using the infinite, isotropic Green’s function, one can apply Table 1: Twenty-seven equations in twenty-seven terms of βkl,h per constituent data point
k 1 1 2 2 3 3 3 2 1 1 1 2 2 3 3 3 1 2 1 1 2 2 3 3 3 2 1
l 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1
k l h 1 2 1 2 1 2 3 3 3 1 3 1 3 1 3 2 2 2 2 3 2 3 2 3 1 1 1
1 1 2 2 1 2
1 3 1 ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Known Terms of βkl,h 1 2 2 2 2 3 3 1 1 3 3 1 2 1 2 1 2 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
3 1 2
3 3 2 2 1 2
1 1 3 ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Unknown Terms 1 2 2 2 3 3 1 2 3 1 3 3 3 3 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
3 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Diagonal Terms 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 2 2 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 3 3
Each row represents an equation generated at each point in the EBSD grid; that is, when equation (6) is evaluated with the specified indices (kl,h). Each column represents a term present in an equation where (kl) are the indices of β and h is the derivative direction. For example, equations generated with (kl,h)=(13,1) would have terms containing β13,1, β13,2, and ten other terms of βkl,h.
equation (6) to a grid of EBSD points to generate a system of equations; the system will have 27 equations per point (as depicted in Table 1). The terms containing a derivative of β in the 3direction are unknowns; the terms based off of the diagonal terms of β are also treated as unknowns but this is mitigated by inclusion of eqns. (9,10) in the system. This system is clearly overconstrained, and decouples into several separate independent systems that are individually less computationally taxing than considering the whole system. For example, the (12,3), (22,3), (32,3), (22,1), (11,2), and (33,2) components can be calculated by solving a system of equations using (kl,h) = (12,1), (12,3), (22,1), (22,3), (32,1), (32,3), (32,2), and (12,2) in connection with eqns. (9,10). Once all derivatives of the infinitesimal elastic distortion tensor are measured or calculated, all components of Nye’s tensor can easily be computed. Summary and Conclusion All components of Nye’s tensor can be recovered if all the partial derivatives of the infinitesimal elastic distortion tensor are known. However, it is not possible to use EBSD to measure derivatives normal to the sample surface. This makes certain components of Nye’s tensor experimentally inaccessible. By combining experimentally obtainable infinitesimal elastic distortion data with underlying field equations, a large, fully-constrained system of linear equations can be constructed that can be solved for the experimentally inaccessible derivatives of elastic distortion. References [1]. A. Arsenlis, D.M. Parks. Crystallographic aspects of geometrically-necessary and statisticallystored dislocation density, Acta Metall. 47 (1999) 1597-1611. [2]. W. Pantleon, Resolving the geometrically necessary dislocation content by conventional electron backscattering diffraction, Scripta Metall. 58 (2008) 994-997. [3]. T. Hardin, B. Adams, D.T. Fullwood, Recovering the Full Dislocation Tensor from HighResolution EBSD Microscopy, Submitted to ICHMM-2011 Conference Proceedings (2011). [4]. S. Villert, C. Maurice, C. Wyon, R. Fortunier, Acuracy assessment of elastic strain measurement by EBSD, J. Microsc. 233 (2009) 290-301. [5]. C.J. Gardner, B.L. Adams, J. Basinger, D.T. Fullwood, EBSD-based continuum dislocation microscopy, Int. J. Plasticity, 26 (2010) 1234-1247. [6]. T. Mura, Micromechanics of defects in solids, 2nd ed., Kluwer Academic, New York, 1987. [7]. C. Teodosiu, Elastic models of crystal defects, Springer, Berlin, 1982. [8]. A.J. Wilkinson, G. Meaden, D.J. Dingley, High-resolution elastic strain measurement from electron backscatter diffraction patterns: new levels of sensitivity, Ultramicroscopy 106 (2006) 307313. Acknowledgement: This work has been funded by a collaborative research grant from the U.S. National Science Foundation: award #s 0936340(OSU) and 0936337(BYU).
