J. Chem. Chem. Eng. 6 (2012) 838-842
Simulation of X-ray Diffraction Line Broadening Caud by Stress Gradients and Determination of Stress Distribution by Fourier Analysis
Vladimir Ivanovitch Monin1*, Joaquim Teixera de Assis1 and Susana Marrero Iglesias2
1. Politechnical Institute of Rio de Janeiro State University, Nova Friburgo-RJ, Brazil
2.Santa Cruz State University, Ilhéus, Brazil
Received: September 10, 2012 / Accepted: September 22, 2012 / Published: September 25, 2012.
Abstract: Different physical, mechanical and chemical process, such as: ion implantation, oxidation, nitridation and others create on the surface of materials residual stress state, characterized by high level and strong gradient. X-ray diffraction method widely ud for stress measurements has some difficulties in interpretation of experimental data, when the depth of X-ray penetration is compared with thickness of surface layer where inhomogeneous stress distribution is localized. Early it has been shown by authors that diffraction line broadening occurs when analyzed surface is characterized by strong gradient. The interest to study the diffraction line broadening is connected to the possibility of ob
taining information about parameters of surface stress distribution. In the prent paper the convolution and deconvolution concepts of Fourier analysis were applied to study X ray diffraction line broadening caud by surface stress gradients. Developed methodology allows determining of stress distribution in superficial layers of materials.
Key words:Stress gradient, X-ray diffraction, computer simulation, Fourier analysis.
1. Introduction
Residual stress arising after modern surface treatments are characterized usually by high level and strong gradient. In this ca analysis of residual stress state by X-ray diffraction method has some difficulties with interpretation of experimental data when the depth of X-ray penetration is compared with thickness of stress distribution with strong gradient.珠子
Computer simulation was ud early to study the influence of surface stress gradients on X-ray diffraction profile [1] that has shown that there are a displacement and broadening of diffraction line caud by strong gradient. The method of this profile simulation is like to the convolution process operating with two mathematical functions to produce the third
*Corresponding author: Vladimir Ivanovitch Monin, Associate Professor, rearch field: X-ray diffraction methods appliedtoanalysisofmaterials.E-mail:***************.function. This fact motivates using of Fourier analysis [2] to study of this kind of diffraction line broadening and apply it to determine stress distribution over the depth of material.
The objectives of prent paper are the following: •Using paration of surface layer on ries of individual layers to form a diffraction profile by sum of individual profiles (direct simulation). Applying this methodology to analyze the influence of surface stress gradient on position and width of diffraction line; •Using functions describing standard profile, absorption of X-ray radiation and stress distribution to develop the methodology of simulation of diffraction line by convolution of the functions (convolution simulation);
•Using process of deconvolution applied to diffraction line broadened by stress gradient to restore the distortion function and then to determine the surface stress distribution.
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Simulation of X-ray Diffraction Line Broadening Caud by Stress Gradients
and Determination of Stress Distribution by Fourier Analysis
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2. Direct Simulation of Diffraction Profile
脑体倒挂Computer simulation is bad on paration of surface on quence of individual layers and account of contribution of the layers to form a total diffraction line. Taking account an attenuation of X-rays into the material on path z for incident and diffracted beams the equations to describe the intensity reflected by any subsurface layer are the following:
I(z) = I 0 e -μz (1)
I dif = a ·I 0 e -μt ·(1/cos(ψ + 90 － θ) +1/cos(ψ － 90
+ θ)) (2) Here I 0 is the intensity of incident beam, μ is the
X-ray absorption coefficient, t is depth of material, ψ is the polar angle, η = 90 － θ, where θ is the diffraction angle. Fig. 1 shows relations between variables parameters for ud functions.
Angular position of diffracted intensity depends on the crystal interplanar distance that is the function of
stress distribution σ(t). For linear distribution the stress on the depth t is expresd by formula σ(t) = σ0
+ kt where σ0 is the stress value on the surface of
material and k is the gradient parameter of stress distribution. If the value of σ0 is normalized to 1 then linear stress distribution can be write as:
1
(3)
The strain εϕψ in arbitrary direction for spherical coordinate system and plane stress state can be expresd [3] by the following equation:
εϕψ = [(1+ν)σ(t ) sin 2ψ/E ] － ν(σ1 + σ2)/E (4) For uniaxial stress state σ1 = σ(t) and σ2 = 0 then Eq. (4) can be written as:
εϕψ = [(1 + ν)sin 2ψ － ν] × σ(t )/E (5) On the other hand with using of Bragg’s law
Fig. 1 The scheme of passage of incident and reflected X-ray beems into the material. deformation εϕψ can be written as [4]:
εϕψ = (d ψ － d 0) d 0 = -ctg θ × (θψ － θ0) (6) where, d 0, d ψ, θ0 and θψ are the interplanar distances and diffraction angles for unstresd and analyzed material, respectively.
Equalizing and simplifying the Eqs. (5) and (6) authors can obtain equation for diffraction angle as a function of depth t and polar angle ψ:
θψ(t ) = θ0 － σ(t) × (sin 2ψ － νcos 2ψ)/E ctg θ0 (7) This equation determines angle position of individual diffraction profiles and other Eq. (2) gives their intensities. Together they are a ba of computer simulation methodology of diffraction profile formed by surface layers of material. Fig. 2 illustrates individual profiles reflected by layers into the surface of material. The sum of the individual profiles forms the total diffraction profile.
Approximating function ud for simulation of individual profiles was modified Cauchy function. The width and doublet distance K α1-α2 for simulated profiles correspond to experimental diffraction line for steel when Cr-K α radiation and (211) reflection are ud.
Fig. 3 shows simulation process for unstresd material or subjected to constant stress.
It can be en that total profile formed by a sum of individual profiles has the same position and width as individual profiles.
Other situation takes place when surface layer is subjected to stress state characterized by strong gr
adient. In this ca individual profiles are displaced from each other and total profile obtained as the sum of individual profiles changes its position and shape.
Fig. 2 Individual profiles reflected by superficial layers of material.
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Simulation of X-ray Diffraction Line Broadening Caud by Stress Gradients
and Determination of Stress Distribution by Fourier Analysis
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Fig. 4 shows simulation of individual and total diffraction profiles for linear stress distribution σ(t) = σ0 + kt , where σ0 and k are equal to 500 MPa and 100 MPa/μm, respectively.
It can be en that stress gradient caus both displacement and broadening of diffraction line proportional to the value of stress gradient.
3. Simulation of Diffraction Profile by Convolution
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3.1 Characteristics of Functions Ud in Convolution Process
Convolution is one of most important concepts in Fourier analysis. It can be defined as a mathematical operation to produce a new function using following equation:
⎰+∞
∞
--=两棵树
τττd )x (f )(g )x (h (8)
Applied to the X-ray diffraction the function g(x) is
Fig. 3 Simulation of diffraction profile for constant stress distribution: (a) profiles formed by individual layers; (b) total profile.
Fig. 4 Simulation of diffraction profile for linear stress distribution: (a) profiles formed by individual layers; (b) total profile (1) and standard profile (2). named as standard function describing distribution of diffracted intensity from perfect crystal; f(x) is usually associated with function of distortion of crystal structure and h(x) is a broadened function describing the diffraction profile of real material.
Possibility of using the convolution concept in simulation of diffraction profile is associated with the fact that simulation process described in paragraph 2.1 is similar to the convolution operation where distortion function depends on two functions. One of them is stress distribution σ(x) and the other is intensity I(x) (Eq. (8)).
Analyzing the influence of the factors the authors can consider that f(x) is the product of the two functions and it can be written as:
f(x) = I(x) × σ(x) (9)
The principal problem of the using of Eqs. (8) and (9) is that the variable coordinate x in Eq. (8) is the angular coordinate of experimental diffraction line and x in Eq. (9) is depth of material. This problem can be resolved by using of Eq. (7) that allows to transform t in θ. Fig. 5 shows graphs of initial functions for diffracted intensity I dif (t) expresd by Eq. (2), normalized stress distribution σ(t)/σ0 and distortion function f(θ) = I(θ) × σ(t)/σ0. Our simulation was made for σ0 = 500 MPA, k equal to 100 MPA/μm and ψ = 00. In this ca linear stress distribution (3) in
normalized form and diffraction angle θψ can be written as:
σ(t)/500 1 0.02 θψ=0 (t ) = θ0 + σ(t) × (ν)/Ectg θ0 Graphs of equations ud for simulation by convolution are shown in Fig. 5.
非攻鲁迅One of the interest facts of using Eq. (9) is the ca when standard function g(x) is Dirac delta function
Fig. 5 Normalized functions (a) I dif (t), (b) σ(t), (c) distortion function f(t) and (d) f(θ) where θ is angular coordinate.
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Simulation of X-ray Diffraction Line Broadening Caud by Stress Gradients
贴的组词and Determination of Stress Distribution by Fourier Analysis
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δ(x). It is known that in this ca convoluted function h(x) has to be identical to distortion function f(x).
On the other hand direct computer simulation carried out with Dirac delta function also has to result a distortion function shown. It is impossible to u in simulation process ideal Dirac delta function but when the standard function shows in Fig. 4b as profile 2, resulting simulated profile 1 (Fig. 4b) has to look like distortion function f(θ) which is shown in Fig. 5d. Fig. 6 prents this ca when computer simulation was made with very sharp standard function. Good coincidence between form of simulated profile (Fig. 6) and distortion function f(θ) (Fig. 5d) can be en.
3.2 Methodology of Simulation by Convolution
Knowing of distortion function f(θ) caud by stress gradient allows to obtain in accordance with Eq. (9) the function of distribution intensity of diffraction profile h(θ) as convolution of this function with standard function g(θ). Fig. 7 shows simulation of h(θ) by using convolution procedure. Parameters and conditions for this simulation were the same as ud in direct simulation prented in Fig. 4.
Fig. 6 Distortion function obtained by direct simulation; two peaks correspond to the components of Kα doublet of diffraction profile.
Fig. 7 Simulation of broadened diffraction profile h(θ) (c) by convolution of standard g(θ) (a) and distortion function f(θ) (b).
It can be en that profile 1 in Fig. 4b has practically the same shape as diffraction profile h(θ) shown in Fig. 7c. Therefore, there were developed two methods of simulation of diffraction profiles. One of them can be named as direct simulation, which is shown in Fig. 4 and the other is simulation by convolution, prented in Fig. 7. Comparison of the two methods of simulation of diffraction profile was made by calculation of integral breadth of diffraction profiles which are shown in Table 1.
Small difference between integral breadths of diffraction lines simulated directly and by convolution confirms correctness of applied distortion function f(θ).
4. Restoring of Stress Distribution by Deconvolution
If h(x) is convolution of g(x) and f(x) functions then Fourier transformations of the functions can be written as:
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H(x) =G(x) × F(x)(10) Here H(x), G(x) and F(x) are the Fourier transforms of functions h(x), g(x) and f(x), respectively.
This equation allows us to find firstly the Fourier transform F(x) as:
F(x) =H(x)/G(x)(11) and then using F(x) restore the unknown distortion function f(x) by deconvolution process. Fig. 8 shows determination of f(x) by deconvolution.
Knowing of distortion function allows finally obtaining of stress distribution over the depth of material.
Table 1 Comparison of integral breadths of diffraction profiles obtained by direct and convolution simulations.
Stress distribution
Integral breadth (conventional units)
Direct simulation
Simulation by
convolution
σ(t) = 500 – 25t8.7 8.6 σ(t) = 500 – 100t12.9 12.0 σ(t) = 500 * e-0.103t9.2 10.1 σ(t) = 0 standard profile8.2 8.2
(a) (b) (c) All Rights Rerved.
Simulation of X-ray Diffraction Line Broadening Caud by Stress Gradients
and Determination of Stress Distribution by Fourier Analysis
842
Fig. 8 Restoring of distortion function by deconvolution: (a) distorted function h(x); (b) standard function g(x); (c) distortion function f(x).
Fig. 9 Restoring of stress distribution function: (a) f(x)-1;
(b) I(x)-2;(c) σ(x).
It was mentioned early (e Eq. (9)) that distortion function f(x) is the product of unknown stress distribution function σ(x) by known attenuation intensity function I(x).
Therefore, stress distribution function σ(x) can be determined by division of f(x) by I(x). Fig. 9 shows this procedure.
It can be en that linear distribution of stress over the depth of material obtained by deconvolution and shown in Fig. 9c corresponds to the function σ(t) ud earlier to simulate distorted function h(x).
5. Conclusions
It has been prented methodology of computer simulation of diffraction profiles that applied to analyze the influence of stress gradients on its position and shape.
It has been developed methodology of determination of stress distribution over the depth of material by using of Fourier analysis.
References
[1]Monin, V. I.; Assis, J. T.; Philippov, S. In Study of Stress
Gradients Using Computer Simulation of Diffraction
Data, Proc. SPIE, Bellingham, USA, 2004; pp 196-199. [2]Bracewell, R. N. The Fourier Transform and Its
Applications; McGraw-Hill: Boston, 2000.
[3]Timoshenko, S. P.; Goodier, N. J. Theory of Elasticity;
McGraw-Hill: New York, 1980.
[4]Hauk, V. Structural and Residual Stress Analysis by
Nondestructive Methods; Elvier: Amsterdam, 1997.
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