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УДК 539.4
A computational scheme for the interaction between an edge dislocation and an arbitrarily shaped inhomogeneity via the numerical equivalent inclusion method
P. Li1, X. Zhang1, D. Lyu1, X. Jin1, and L.M. Keer2
1 State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400030, China 2 Department of Mechanical Engineering, Northwestern University, Evanston, Illinois, 60208, USA
The interactions between dislocations and inhomogeneity may play an important role in strengthening and hardening of materials. The problem can be solved analytically only for limited cases of simple geometry. By employing the recently developed numerical equivalent inclusion method, this work presents an effective computational scheme for studying the stress field due to an edge dislocation in the vicinity of an arbitrarily shaped inhomogeneity. The inhomogeneity is treated as an equivalent inclusion that is numerically discretized by rectangular elements. The mismatch between the matrix and the inhomogeneity materials are formulated through Dundurs' parameters for numerical stability and robustness. The proposed method can efficiently and accurately evaluate the elastic field of the equivalent inclusion with the assistance of a fast Fourier transform based algorithm, constituting an essential refinement of the existing approach in the dislocation-inhomogeneity literature. Several benchmark examples are examined to demonstrate the flexibility, efficiency and accuracy of the present method.
Keywords: edge dislocation, arbitrarily inhomogeneity, Dundurs' parameters, numerical equivalent inclusion method, fast Fourier transforms
DOI 10.24411/1683-805X-2018-16017
Расчет взаимодействия краевой дислокации с неоднородностью произвольной формы с использованием метода эквивалентного включения
P. Li1, X. Zhang1, D. Lyu1, X. Jin1, L.M. Keer2
1 Чунцинский университет, Чунцин, 400030, КНР 2 Северо-Западный университет, Эванстон, Иллинойс, 60208, США
Взаимодействие дислокаций с неоднородностью может играть важную роль в упрочнении материалов. Данная задача имеет аналитическое решение только для некоторыж случаев простой геометрии. В работе на основе численного метода эквивалентного включения предложена эффективная выгшслительная схема для исследования поля напряжений, создаваемого краевой дислокацией вблизи неоднородности произвольной формы. Неоднородность рассматривается как эквивалентное включение, которое в модели описывается прямоугольными элементами. Для обеспечения устойчивости модели различие материалов матрицы и включения учитывается через параметры Дундурса. Предложенный метод позволяет эффективно и точно оценить упругое поле эквивалентного включения с помощью алгоритма на основе быстрого преобразования Фурье и существенно уточнить традиционный подход к рассмотрению дислокаций и неоднородностей. На примерах показана адаптивность, эффективность и точность предлагаемого метода.
Ключевые слова: краевая дислокация, произвольная неоднородность, параметры Дундурса, численный метод эквивалентного включения, быстрые преобразования Фурье
1. Introduction
Barenblatt's seminal works [1, 2] proposed a cohesive zone model to study the fracture and failure behavior in brittle materials. Based on the hypothesis of "cohesive forces", the singularity at the crack tip is circumvented and
the stability of crack propagation is discussed [2]. From the materials' point of view, crack mechanism usually involves a dislocation pile-up. Since the mobility of the dislocation can be remarkably affected by the internal force arising from the inhomogeneity, the crack propagation may
© Li P., Zhang X., Lyu D., Jin X., Keer L.M., 2018
be significantly hindered or promoted in the presence of material inhomogeneities, especially when the inhomoge-neities are located in the vicinity of the crack tip [3-6].
The interaction of dislocations with the inhomogeneities is a crucial mechanism for the strengthening and hardening of materials [7-9]. In continuum mechanics, a crack problem may be solved by the distributed dislocation technique, which is applicable to the crack-inhomogeneity interactions, where the solution of the dislocation-inhomo-geneity is treated as a Green's function [10, 11].
Dundurs and Mura [12] first presented a closed-form solution for the elastic field of an edge dislocation located outside a circular inhomogeneity, and some related work on circular inhomogeneity were subsequently conducted [13, 14]. Stagni and Lizzio [15] analytically solved the interaction between an edge dislocation and an elliptical in-homogeneity. Their solution was represented by a Laurent series expansion. Hills et al. [16] has documented the available classical dislocation solutions in the literature. It is seen, however, that the analytical works on the interactions between dislocation and inhomogeneity are limited to specific and usually simple geometries.
Hutchinson [17] developed an approximate method to study the shielding effect of profuse microcracking at the tip of a macroscopic crack. In his work, the effects of the microcracks relieving the residual stress are manifest on the macroscopic scale as transformation strains. The method is regarded as the zeroth-order approximation [18] in that the transformation strain is determined by the far field loading only, while the local disturbances are neglected. Hutchison approximate method was later employed by Li and his co-workers [19-21] to study the interactions between crack/dislocation and inhomogeneities.
The equivalent inclusion method was originally proposed by Eshelby [22] for solving three-dimensional (3D) ellipsoidal inhomogeneity problem. In the context of micro-mechanics [10], the subdomain containing eigenstrain (i.e., a generic name for non-elastic strains) but having identical material is called an "inclusion", to distinguish the term "inhomogeneity," which refers to a subdomain with different material from its surrounding matrix. Jin et al. [23] demonstrated that equivalent inclusion method is also a handy and elegant tool for analyzing 2D inhomogeneity in plane elasticity. Zhou et al. [18] developed a numerical approach based on the equivalent inclusion method for handling arbitrarily shaped inhomogeneities of any material constants. The method, termed as "numerical equivalent inclusion method", employs Dundurs' parameters for describing any mismatch combinations between the inhomogeneity and the matrix materials, and has been demonstrated to be robust and stable over the entire Dundurs plane. In the implementation of numerical equivalent inclusion method, the equivalent inclusion domain is discretized into a system of elementary inclusions, each being subjected to uniform eigen-
strains [10] whose magnitude is to be determined through an iterative scheme. The numerical equivalent inclusion method shows a refinement over the well-known Hutchison's approximate method, because the latter may be viewed as neglecting a coupling term and hence the iteration revoked in determining the unknown eigenstrains.
Even for homogeneous materials, the classical dislocation solution contains a Cauchy type singularity [16], which usually is difficult to be handled by commercial finite element software. In the presence of inhomogeneities, the disturbance incurred may be viewed analogously from an inclusion with properly chosen eigenstrains, following the renowned Eshelby's equivalent inclusion method. It is therefore envisaged that the interactions between dislocations and inhomogeneities may be effectively solved via the numerical equivalent inclusion method.
The paper is organized as follows. In Sect. 2, a pictorial explanation of the equivalent inclusion method, as well as solution of an edge dislocation in a 2D homogenous medium are reviewed first. The elementary solution of a rectangular inclusion is recorded and is then utilized as building blocks for analyzing any arbitrarily shaped inclusion. Furthermore, the implementation of the numerical equivalent inclusion method is presented, where the iterative scheme for determining the equivalent eigenstrain distribution is discussed in detail. In Sect. 3, benchmark examples are utilized to validate the effectiveness of the present solution and the parametric studies on the effects of mesh size, iterative number as well as the aspect ratio are investigated. The main concluding remarks are summarized in Sect. 4, and the detailed expressions for stresses produced by a rectangular inclusion subjected to uniform eigenstrains are given in the appendix.
2. Formulation
Consider an infinite plane containing an arbitrarily shaped inhomogeneity. An edge dislocation is located outside the inhomogeneity, which is assumed to be perfectly bonded to the matrix. As shown in Fig. 1, the infinitely
x2 j
Fig. 1. Edge dislocation in the neighborhood of an arbitrarily shaped inhomogeneity
corresponding inclusion (b)
extended matrix denoted by region 1 has elastic moduli Cijkl, and the subdomain (inhomogeneity) with elastic moduli C*kl is designated as region 2. Hereafter, the subscripts 1 and 2 are also adopted in the context to distinguish whether the quantity is associated with the matrix or inhomogeneity.
In view of the equivalent inclusion method [23], the problem in Fig. 1 may be solved by superposing a homogeneous material solution (Fig. 2, a) and a corresponding inclusion solution (Fig. 2, b). The former subproblem (Fig. 2, a) considers an edge dislocation in the absence of the inhomogeneity, and the classical 2D solution will be recorded in Sect. 2.1. The latter subproblem (Fig. 2, b) conceives an equivalent inclusion for the disturbance due to the existence of the inhomogeneity. Since in the current work the inhomogeneity can be any arbitrary shape, the corresponding inclusion in Fig. 2, b will be numerically discretized into a number of elementary inclusions, whose formulation will be discussed in Sect. 2.2. The numerical implementation of the equivalent inclusion method will be elaborated in Sect. 2.3, where an effective iterative scheme is presented for determining the distribution of the eigenstrains in the equivalent inclusion.
It has been shown in [23] that the equivalent inclusion method formulation for plane stress is apparently simpler, because the longitudinal eigenstrain component will not vanish in the case of plane strain, unless Poisson's ratios of the inhomogeneity and the matrix are the same value or the inhomogeneity degenerates to a cavity. For simplicity, only plane stress problem is investigated in the current numerical study. However, it is understood that the solution of plane stress may be readily utilized to interpret the case of plane strain, with the conversion of Kolosov's constant [24].
2.1. Edge dislocation solution in an infinite homogeneous plane
The classical solution of an edge dislocation in an infinite homogeneous medium (Fig. 2, a) has been documented in Ref. [16]. In the Cartesian coordinate system, the edge dislocation is located at (x', x'2) and has Burgers vector
components b1? b2. The induced stresses at a field point (x1? x2) is given by [16]:
Co
C22
C o
C>12
n(K1 +1)
111
122
112
211
222
212
(1)
where the superscript "0" indicates the homogeneous material solution, Kolosov's constant k = (3 -v)/(1 + v) for plane stress, with v being Poisson's ratio, ^ is the shear modulus, the influence coefficients Gtjk are
% 2(3^?+% 2)
^111
G112 =
G222 _
G
r
&2
122
.% 2 (%2-% 2)
%1( %2 -%2)
r
n2 e2>
G211 =
%1 -%2)
(2)
%,(%2+3%2)
G212 _
% 2(%2 -% 2)
4 ' 212 _ 4 r r
r r
in which %1 = x1 - x[, % 2 = x2 - x2, and r = ^J%? + %2. Note that the first subscript of the above components is associated with Burgers vector and the remaining two indices are related to the stress components. It is well-known that the above dislocation influence functions contain a singularity when the core of the dislocation is approached. The singularity may lead to numerical issues when a dislocation problem is solved by the commercial finite element software.
2.2. Elementary solution for a rectangular inclusion
Since the inclusion can be any arbitrary shape and in order to take advantage of the fast Fourier transform based algorithms [25], the computational domain that encloses the inclusion is discretized into a number of uniform rectangular elements (Fig. 3). The element size is sufficiently small so that the equivalent eigenstrain is assumed to be uniform inside each rectangular patch. A typical rectangular element having dimensions A1 xA2 and centered at (x[0, x20) is shown in Fig. 3. The rectangular inclusion, whose sides are parallel to the coordinate axes, is subjected to uniform eigenstrains e-. The stresses inside and outside the rectangular inclusion may be represented in a unified
Rectangular ^element
Computational domain
Fig. 3. Schematic of the discretization for an arbitrarily shaped inhomogeneity with uniform rectangular elements (color online)
expression [26]:
22
12
1
4n(1 — Vi )
^1111 T122 T11
12
*{K) \ n yf / 0 I Determine eigenstress K = K+ 1
(3)
where Ttjkl is the elementary influence coefficient and are detailed in the Appendix.
With the superposition principle, the resultant elastic field due to the original arbitrarily shaped inclusion (cf. left of Fig. 3) is evaluated by summing contributions from each element. It should be emphasized that a direct superposition may lead to a huge computational burden, especially for a fine mesh. However, the summation has a convolution nature when the computational domain is dis-cretized into uniform rectangular meshes, and hence can be tremendously accelerated by taking advantage of the fast Fourier transform related algorithms [25].
2.3. Implementation of the equivalent inclusion method
The fundamentals of equivalent inclusion method have been detailed in [23]. The essence of equivalent inclusion method is established upon the consistency condition that the elastic field disturbance due to the presence of inhomo-geneity can be simulated by the resulting field generated by a corresponding inclusion when the eigenstrain is chosen properly. That is, according to Eshelby [22], the consistency condition is enforced inside the inclusion:
Cijkl (8k + Sklmn8 mn ) = Cijkl (8kl + Sklmn8mn - 8kl)' (4)
where 80l is determined from the homogeneous material solution (cf., Fig. 2, a), and Sklmn is the Eshelby tensor of the inclusion. Theoretically, the unknown eigenstrain 8 ^ is obtained after solving the complex tensorial equation.
Alternatively, it may be more convenient to rewrite Eq. (4) in terms of compliance moduli Mjkl = Cjlu. For an isotropic material with Young's modulus E and Poisson's ratio v, Mjkl = [M jl + 5i75jk )(1 + v )/2-vSjS« ]/E. The details have been discussed in [23], and the counterpart consistency condition is derived as
Mtjki where a
(a° + a k) = Mjki (a°ki
ijklV^kl + akl) + 8 ij, (5)
kl is the eigenstresses produced by the unknown eigenstrains (cf. Fig. 2, b). As discussed in Sect. 2.2, the eigenstresses may be solved numerically for a given distribution of eigenstrains (Fig. 3). By defining AMjkl as the
compliance moduli difference between the inhomogeneity
*
and the matrix material, i.e. AMijkl = Mijkl - Mjjkl, one may construct an iteration formula as follows:
= AMijki
(a 0i
+ a ki )•
(6)
The above Eq. (6) is applicable for a general 3D configuration. In the present numerical implementation of the plane stress problem, only the in-plane components are retained, and the compliance moduli difference may be presented in matrix form as
[Mi ]=
K +1
4^1 (1 + a)
where
A m
1111
A m
1122
Am
2211
Am
2222
0 0
Am1212
Am1 m =Am2222 =a
Am1122 =Am2211 = 2P —a ,
Am1212 = 4(a —P), and Dundurs' parameters a and P are ^ 2( K +1) — k 2 +1) ^2 ( K +1) + K 2 + 1)' ^ 2 ( K1 —1) — ^1(K 2 — 1)
(7)
(8)
a :
P
(9)
^ 2( Kl + 1) + K 2 + 1) It is also noted that in Hutchinson's approximation, the disturbance term (eigenstresses) akl at the right hand side of is neglected, hence the equivalent eigenstrain components 8 ij are directly estimated by AMjkla°kl. Since no iteration is involved, the approximation is called "zeroth order". However, such simple estimation may result in relative rough approximation.
f Input K = 0, Gy = 0 /
*(K) \ j[ /f / 0 I (K)\ 8 Ù = Mdytffaij + a y )
_ Determine eigenstress
K = K+ 1
Fig. 4. Flow chart of the numerical equivalent inclusion method
Fig. 5. An edge dislocation interacting with an elliptical inhomo-geneity in an infinite elastic medium
The present computational scheme may start from Hutchinson's approximation, but with succeeding refinement through iteration, as shown in the flow chart (Fig. 4). For ease of presentation, the superscript (K) in the flow chart denotes the Kth iteration step. Before the start of iteration, the eigenstresses are assumed to be zero. On each step of iteration, Eq. (6) is used to update the current estimation of the equivalent eigenstrains. Then, the current relative error is estimated. Provided that either the result meets the given accuracy constraints or the maximum allowed number of iteration has achieved, the iteration process is stopped. Otherwise, another iteration is performed. In this case, the eigenstresses are re-evaluated using the current eigenstrains (cf. Fig. 3), and then Eq. (6) is recalled to update the equivalent eigenstrains.
3. Results and discussions
The present computational scheme is programmed in standard FORTRAN language and implemented in double precision. The computations are performed on a personal computer with a 3.4 GHz i7 CPU and 16.0 GB memory.
In order to illustrate the application of the aforementioned method, an edge dislocation interacting with an elliptical inhomogeneity in an infinite elastic matrix is considered (Fig. 5). The edge dislocation is located at (2a1, 0) and has Burgers vectors (b1, b2). The inhomogeneity is centered at the origin of Cartesian coordinate systems and has semi-axes a1 and a2. The main components of the computational parameters are listed in table.
Table
Computational parameters of the inhomogeneity and the edge dislocation
Parameters
Young's modulus £1, E2, GPa
Poisson's ratio vl9 V2
Burgers vectors (b1, b2)
Values
200, 300
0.3, 0.2
Fig. 6. Comparisons of the normalized shear stress for Dundurs-Mura solution and present solution
The closed-form solution obtained by Dundurs and Mura [12], corresponding to a circular inhomogeneity (aspect ratio a2j a1 = 1), is utilized to validate the present solution. In the benchmark study, the mesh is chosen as 128 x 128 and the maximum allowed number of iteration is set to be 5. Comparisons of the normalized shear stress along the axis x1 are shown in Fig. 6. It is demonstrated that the present numerical equivalent inclusion method solution exhibits excellent agreement with the exact solution.
In order to better illustrate the disturbance due to the presence of the inhomogeneity, the singular Cauchy term is then removed, and the normalized shear stress component along the axis x1 is compared with the zeroth-order results by Hutchinson approximation [17]. As illustrated in Fig. 7, the present numerical equivalent inclusion method solution consistently shows better agreement with Dundurs-Mura solution [12]. More related details are also reported later in Fig. 8, where 0th approximation refers to the results by Hutchinson's estimation.
One of the key issues in numerical equivalent inclusion method is the discretization error along the geometrical boundaries, since uniform rectangular mesh has to be used in order to take advantage of fast Fourier transform accel-
(1.0, 0.0)
Fig. 7. Comparisons of the normalized disturbance shear stress for Dundurs-Mura solution, Hutchinson solution and present solution
Fig. 8. Average error of different approximations for an edge dislocation interacting with a circular inhomogeneity
eration. To address this issue, results of different mesh schemes are investigated. As illustrated in Fig. 9, typical results for meshes of 32x32 and 512x512 are compared with the previous one using 128 x 128. The present computation shows that a finer discretization usually yields better accuracy. However, even the coarsest mesh of32x32 shows satisfactory agreement with the exact solution.
Numerical tests are also performed extensive to illustrate the CPU time savings by virtue of the fast Fourier transform techniques. Comparisons with time cost by the direct superposition are shown in Fig. 10, where the exact time of direct superposition for the mesh of 512x512 is excluded. When performing the superposition in Fig. 3, the CPU time are mainly consumed by the multiplication operations. Assuming the computational domain is dis-cretized to Nx x Ny rectangular patches, O(NXN^) multiplication operations are needed for the method based on direct summation. However, the time for the method utilizing the fast Fourier transform algorithms is estimated to be O(NxNy ln(NxNy)) multiplication operations. Therefore, when the 2D grids are refined by a factor of 2 in both directions, the time consumed by the direct superposition is expected to increase by a factor of 16, as opposed to 4 when
Fig. 10. Comparisons between the direct superposition and fast Fourier transform (FFT) algorithms in terms of computational time cost
fast Fourier transform algorithms are adopted. The above CPU time estimation roughly agrees with our numerical executions (Fig. 10). Note that for a large number of elements, the memory access may also consume considerable time, which could be the cause of the small amount deviations from the CPU time estimation.
Moreover, the dependence of the numerical accuracy on the mesh size as well as the iteration number is investigated (Fig. 8). The average relative error of the stresses 8(a) in this work is defined as
1 Nk
8(a)=N-?
Nk i =1
a -a
ae
x 100%,
(10)
where Nk, ae and aa are the number of the field points, the exact value and the approximate numerical solution, respectively. As demonstrated in Fig. 8, the average error for zeroth order approximation may exceed 20%, and would be significantly alleviated after the first iteration. After the 5th iteration, the average relative errors in most cases drop below 5%, sufficient for engineering applications. For a
Fig. 9. Effects of mesh size for the normalized disturbance shear stress
Fig. 11. Effects of aspect ratio a2/a1 = 0.1 (1), 0.5 (2), 1.0 (5), 2.0 (4), 10.0 (5) for the normalized disturbance shear stress
fixed iteration number, the numerical error drops rapidly from the mesh of 32x32 to 128x128, and then varies moderately as the discretization becomes even finer.
The effects of the aspect ratio a2/aj for the normalized shear stress along the axis xj is also investigated in Fig. 11. For the cases of 0.1, 0.5 and 1.0, the dimensionless results decline gradually and then tend toward a stable value as xj /aj ranges from 1.5 to 2.5, while the normalized disturbance shear stresses for the other cases would rise to a peak and then decrease slightly. It can be seen that the shape effects of the inhomogeneity may have a significant effect on the interaction.
4. Concluding remarks
In reality, the inhomogeneities in the engineering materials are not only of arbitrary shape but also distribute in a random way. The interaction between edge dislocations and inhomogeneities may have a significant influence on the crack growth in heterogeneous materials and strain hardening in metal alloys. However, analytical solutions only exist for limited cases. The involved singularity nature may even cause numerical obstacles when solved by the commercial finite element software. The recently developed numerical equivalent inclusion method may a choice for quantitative analysis of the dislocation-inhomogeneity interactions. Satisfactory accuracy is achieved by an iterative scheme when compared with Hutchinson's approximate method.
In this work, the fast Fourier transform based algorithms can be seamlessly incorporated with the numerical equivalent inclusion method. Dundurs' parameters are employed in the present equivalent inclusion method formulation to enhance the numerical stability and robustness. The effectiveness of the iterative method is illustrated by comparison with the analytical solution. The parametric studies demonstrate that the present method may considerably improve the computational efficiency and accuracy. Although only the circular or elliptical inhomogeneity is exemplified in this work, the discrete rectangular elements can be treated as building blocks, and therefore it is anticipated that the present numerical equivalent inclusion method would be flexible and versatile for analyzing inhomogeneities of arbitrary shape.
Acknowledgments
This work is supported by Fundamental Research Funds for the Central Universities (2018CDYJSY0055). P. Li and X. Zhang are grateful to the Graduate Research and Innovation Foundation of Chongqing, China (Grant Nos. CYB18020 and CYB17025). X. Jin would acknowledge the support from Fundamental Research Funds for the Central Universities (106112017CDJQJ328839), the National Natural Science Foundation of China (Grant Nos. 51475057 and 51875059), and the State Key Laboratory of Mechani-
cal Transmissions through funding (SKLMT-ZZKT-2017M15).
Appendix
The elementary influence coefficient Tijkd in Eq. (3) may be formulated as
2 2
Tijki = 1) ji (x1a, X2P ),
a=1 P=1
where x1a, x2p (a, P = 1, 2) are
J Xn = Xj — Xjo —Aj/2, x21 = x2 — x20 —A2/2,
[x12 = X1 — x10 + A1/2, x22 = x2 — x20 + A2/2-The detailed expressions for five necessary functions
j(£^ £2) are
— 4arctan ^
(A1)
(A2)
1111 ^ +£2 -......£2
_ 2£1£2
¿1122 = ¿1212 £2 + £2 ' £1 +£2
¿1112 = M£2 + Ç22)—A' £1 + £2
¿1222 = ln (£2 + £2) — tÎ+L £1 +£2
(A3) (A4) (A5) (A6)
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Received October 15, 2018, revised October 15, 2018, accepted October 22, 2018
Ceedeuua 06 aemopax
Pu Li, PhD candidate, Chongqing University, China, [email protected] Xiangning Zhang, PhD candidate, Chongqing University, China, [email protected] Ding Lyu, PhD candidate, Chongqing University, China, [email protected] Xiaoqing Jin, Prof., Chongqing University, China, [email protected]
Leon M. Keer, Professor Emeritus, Northwestern University, USA, [email protected]
