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The elastic fields of misfit cylindrical particles: a dislocation-based numerical approach
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 30 (2003) 325–334 www.elsevier.com/locate/mechrescom
The elastic fields of misfit cylindrical particles: a dislocation-based numerical approach Jesus Lerma a, Tariq Khraishi b
a,*
, Yu-Lin Shen a, Brian D. Wirth
b
a Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA Materials Science and Technology Division, Lawrence Livermore National Laboratory, Livermore, California 94551, USA
Abstract Analytical solutions for the displacement and stress fields due to a misfit particle in a host matrix can be difficult to obtain, especially for complex particle geometries. In this work, we present a numerical method for finding such fields in the case of infinitely-long particles. The method is based on discretizing the continuous misfit region between the particle and matrix into local misfit regions consisting of interstitial dislocation loops. The results presented here indicate very good agreement with analytical solutions and better convergence with increasing loop density. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Dislocation mechanics; Eigenstrains; Particle strengthening
1. Introduction The problem is depicted in Fig. 1, which shows the cross-section of an infinitely-long misfit particle, of radius r0 , placed into a smaller size hole within an infinite matrix. The radial misfit between the particle and the hole is denoted as d. In this figure, the particle and hole are shown to be perfect cylinders with uniform d along the interface. In general this need not be the case and is only shown here for ease of illustration of the present method. Also, the elastic constants of the particle can be different from those of the matrix. In micromechanics, it is desirable to quantify the displacement and stress fields, in the particle and matrix materials, caused by the presence of such misfit particles. Ideally, the solution is sought for any particle geometry or shape and any distribution of the misfit along the interface. Analytical solutions are typically derived for simplified particle geometries as well as for well-described or behaved misfit distributions. For example, Teodosiu (1982) solved the boundary-value problem (BVP) of a spherical misfit particle of uniform radial misfit and of different elastic properties than those of the matrix. Below we concisely follow TeodosiuÕs approach to give the solution in the case of a cylindrical particle, i.e. the 2D problem. This solution is provided for completeness and to compare with the numerical method presented in this paper. The main disadvantage of the BVP method is that every time the geometry of the problem changes, the
*
Corresponding author. Tel.: +1-505-277-6803; fax: +1-505-277-1571. E-mail address: [email protected] (T. Khraishi).
0093-6413/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00035-1
326
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
δ
r0
R
G',v' G,v
Fig. 1. A cylindrical particle of radius r0 inserted into a smaller hole in a cylindrical shell/matrix of radius R. The amount of misfit between the particle and the matrix is denoted by d. The particle-matrix interface can be approximated with a polygonal prism like the the hexagon shown in the figure.
solution must be recreated. Of course, the analytical solution for an infinite cylindrical particle with constant or uniform misfit can also be deduced as a special case from more advanced analytical solutions. For example, it can be deduced from the solution for an ellipsoidal particle presented by Eshelby (1957), among others, using the eigenstrain method. Mura (1982) presents an excellent treatise of the eigenstrain method for solving particle problems, including misfit situations. However, once again, every time the problem geometry changes, the solution must be re-formulated, and the eigenstrains are typically restricted to simple analytical functions. Below, we present a distributed-dislocation method for numerically solving particle misfit problems of any geometry and any misfit distribution along the interface. Here the state of continuous misfit region along the particle-matrix interface is replaced by a distribution of local misfits. Each local misfit region is represented by a prismatic interstitial dislocation loop. As the dislocation loop density increases, i.e. the number of loops increase and the size of each decrease, the numerical solution converges to the analytical solution. Here the Burgers vector of each loop corresponds to the local amount of misfit between the particle and the matrix. Due to its numerical nature, the method can be used to quantify the elastic fields of a particle of any geometry and any misfit distribution.
2. Method Fig. 1 shows the cross-section of an infinitely-long elastic cylindrical particle inserted into a smaller cylindrical hole of an elastic shell. The shell radius is denoted by R. The hole radius is smaller than the particle radius r0 by a radial misfit d (or equivalently by a misfit volume dt0 ¼ dð2pr0 Þ per unit length along the particle). In order to analytically solve for the elastic fields induced by this misfit, one has to solve an elasticity BVP subject to certain boundary conditions. Due to the cylindrical symmetry of the deformation field with only a radial component of displacement of relevance, we here use a semi-inverse method starting with a functional form for the displacement: ur ¼ u ¼ uðrÞ;
uh ¼ uz ¼ 0
ð1Þ
Notice that the displacement is independent of the angular position h of a material point P located at ðr; hÞ or equivalently at ðx; yÞ. Substituting Eq. (1) into the typical definition of strain in cylindrical coordinates, one obtains: err ¼
ou ; or
u ehh ¼ ; r
ezz ¼ erh ¼ ehz ¼ ezr ¼ 0
ð2Þ
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
327
Assuming linear elasticity and isotropic particle and shell/matrix materials, HookeÕs law applies and one can find the stresses directly from Eq. (2), i.e. in terms of displacement. The stresses, however, must satisfy the static equations of equilibrium. In the absence of body forces, two of three equilibrium equations are identically satisfied or null. The third, after some algebraic simplifications, can be expressed as: drrr 1 þ ðrrr rhh Þ ¼ 0 r dr
ð3Þ
Substituting the stresses, in terms of displacement, into Eq. (3), one obtains: d2 u 1 du 1 u¼0 þ dr2 r dr r2
ð4Þ
Eq. (4) is a standard Cauchy–Euler differential equation (Kreyszig, 1993) and its solution has the general form: u ¼ uðrÞ ¼ C1 r þ C2 r1
ð5Þ
where C1 and C2 are integration constants. It is to be noted that all of the above equations apply to both the particle and the surrounding matrix. In order to distinguish between the two solutions we will hereafter add a prime to the integration constants and other field quantities belonging to the particle. The solutions for the particle and the matrix are then represented by Eqs. (6) and (7), respectively. u0 ¼ u0 ðrÞ ¼ C10 r þ C20 r1
ð6Þ
u ¼ uðrÞ ¼ C1 r þ C2 r1
ð7Þ
C20
must be equal to zero otherwise the displacement at the center of the particle In Eq. (6), the constant would be infinite. Eq. (6) is then reduced to: u0 ¼ u0 ðrÞ ¼ C10 r
ð8Þ
To solve for C1 , C2 , and C10 we need to apply boundary conditions (BCs). The pertinent BCs are: rrr ðr ¼ RÞ ¼ rrh ðr ¼ RÞ ¼ rrz ðr ¼ RÞ ¼ 0
ð9aÞ
rrr ðr ¼ r0 Þ ¼ r0rr ðr ¼ r0 Þ
ð9bÞ
2pr0 ½uðr ¼ r0 Þ u0 ðr ¼ r0 Þ ¼ dt0
ð9cÞ
With the aid of Eqs. (7–9), (2), and HookeÕs law we solve for the three constants (details skipped): C1 ¼
G C2 þ GÞ
R2 ðk
2 1 dt0 G r02 G r0 1þ 1 k þ G R2 k0 þ G0 R2 2p G 1 1 C10 ¼ 0 C2 k þ G0 R2 r02 C2 ¼
ð10aÞ
ð10bÞ
ð10cÞ
Eq. (10) and above can be used to explicitly solve for the displacement, strain, and stress fields. Note that in Eq. (10), G and G0 stand for the shear modulus of the matrix and particle, respectively; k and k0 stand for LameÕs constant (note that k and G are also both called LameÕs constants). The expressions of Eq. (10) are for the case of a cylindrical particle inside a cylindrical shell of finite radius, where the material properties of the shell and the particle are distinct. In our numerical illustrations below, we specifically deal with the case
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where the media surrounding the particle is infinite, i.e. R equals infinity, and where the particle and the surrounding media possess equivalent material properties. Modifying Eq. (10) for this particular problem we obtain: C1 ¼ 0
ð11aÞ
1 dt0 G 1þ 0 C2 ¼ 2p k þ G0
ð11bÞ
G C2 k0 þ G0 r02
ð11cÞ
C10 ¼
Eq. (11) can be utilized to solve for stresses, strains, and displacements. To illustrate the present numerical method, we consider for simplicity the case where G ¼ G0 , k ¼ k0 , and thus m ¼ m0 (where m is PoissonÕs ratio). The misfit region between the particle and the matrix is approximated with an enclosed polygon (e.g. see Fig. 1). Each side of the polygon stands for a local misfit region. This local misfit representation is facilitated by considering the face of each side to be a prismatic dislocation loop (i.e., one whose Burgers vector b is pointed normal to its habit plane). The dislocation loop in this case is rectangular and is composed of four edge dislocation segments (two of them at z 1 Fig. 2). A prismatic dislocation loop is simply a layer of matter inserted or removed in a surrounding matrix (Hirth and Lothe, 1982; Hull and Bacon, 1984). Fig. 2 shows an oblique view of a section of the enclosed polygon. Each side has a Burgers vector normal to its face. The dislocation segments at the end of the infinite prism do not contribute to the state of elastic deformation in the material. Fig. 3 shows a cross-section of the infinitely-long prism (normal to its axis), with each prismatic dislocation loop being represented by two oppositely-signed, infinite edge dislocations. The elastic fields (displacement, strain and stress) at a field point P ðx; yÞ are numerically found to be, from the principle of linear superposition, the summation of the elastic fields due to the edge dislocations in Fig. 3. Note here that the Burgers vector of each dislocation i, bi , can be decomposed into two components: a glide component bix , and a climb component biy (the terminology of glide and climb components has been
b =δ Fig. 2. A section of the prism in Fig. 1. Each of the surfaces or sides stand for a prismatic dislocation loop, i.e. whose Burgers vector is normal to its plane. The magnitude of the Burgers vector, b, is equal to the misfit between the particle and the matrix d.
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p Edge Dislocation
y
Dislocation Line Sense
Fig. 3. A cross-section of Fig. 2. Each dislocation loop is represented by two oppositely-signed, infinite edge dislocations. Point P is a material point that can reside inside or outside the particle. Also shown is a dislocation i whose Burgers vector is decomposed into x and y components.
borrowed from Weertman, 1996). As explained above, the displacements (ux and uy ) and stresses (rxx , rxy , and ryy ) fields at a material or field point will then be given by: ux ¼
2N X
uix ;
uy ¼
i¼1
rxx ¼
2N X
uiy
ð12Þ
i¼1
2N X
rixx ;
rxy ¼
i¼1
2N X i¼1
rixy ;
ryy ¼
2N X i¼1
riyy ;
rzz ¼
2N X
rizz
ð13Þ
i¼1
Notice that for N sides of the polygon, 2N edge dislocations exist (see Fig. 3) and hence the 2N summation limit in (12) and (13). The stresses and displacements at ðx; yÞ due to a single infinite edge dislocation i located at ðxi ; y i Þ with a Burgers vector bi ¼ bix þ biy are given by (Weertman, 1996): rixx ¼
2 2 2 2 Gbiy Gbix ðy y i Þð3ðx xi Þ þ ðy y i Þ Þ ðx xi Þððx xi Þ ðy y i Þ Þ þ 2 2 2 2 2 2 2pð1 mÞ 2pð1 mÞ ððx xi Þ þ ðy y i Þ Þ ððx xi Þ þ ðy y i Þ Þ
ð14aÞ
riyy ¼
2 2 2 2 Gbiy Gbix ðy y i Þððx xi Þ ðy y i Þ Þ ðx xi Þð3ðy y i Þ þ ðx xi Þ Þ þ 2pð1 mÞ 2pð1 mÞ ððx xi Þ2 þ ðy y i Þ2 Þ2 ððx xi Þ2 þ ðy y i Þ2 Þ2
ð14bÞ
rixy ¼
2 2 2 2 Gbiy Gbix ðx xi Þððx xi Þ ðy y i Þ Þ ðy y i Þððx xi Þ ðy y i Þ Þ þ 2 2 2 2 2 2 2pð1 mÞ 2pð1 mÞ ððx xi Þ þ ðy y i Þ Þ ððx xi Þ þ ðy y i Þ Þ
ð14cÞ
rizz ¼ mðrixx þ riyy Þ
ð14dÞ
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# bix y yi 1 ðx xi Þðy y i Þ 1 þ tan ¼ 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 2p x xi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 i 2 ðx xi Þ2 þ ðy y i Þ2 biy ð1 2mÞ 1 ðy y Þ 5 ln þ þ 4 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 rc 2p 2ð1 mÞ "
uix
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 i Þ2 þ ðy y i Þ2 i i 2 ðx x b 1 2m 1 ðy y Þ 5 uiy ¼ x 4 ln þ 2ð1 mÞ 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 rc 2p " # biy y yi 1 ðx xi Þðy y i Þ 1 : tan þ 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 2p x xi
ð15aÞ
ð15bÞ
In the above equations, rc is a cut-off or dislocation core radius taken here equal to the magnitude of one Burgers vector, b, in our numerical examples presented below.
3. Results and discussion We now illustrate the results of the above method. In Fig. 4, rxx is plotted along the x-axis for m ¼ 0:3 and r0 ¼ 100b. Note that along the x-axis, rrr corresponds to rxx , and rhh corresponds to ryy . The figure shows different curves each corresponding to a given dislocation loop density (the loop densities used are 4, 5, 6, 8 10, 15, 20, 30, and 40 loops, represented by the increasing dash size in Figs. 4–11). As the loop density increases, so does the dash size in the curves. We note in Fig. 4 that, for any given loop density, as the distance from the particle increases the numerical solution approaches the analytical solution (in solid line). For points relatively close to the particle (i.e. at x ¼ 100b), the numerical solution is only accurate for relatively large dislocation densities. Overall, the numerical solution converges to the analytical one as the loop (or mesh) density increases. Figs. 5 and 6 show similar trends, as above, for the cases ofryy and rzz , respectively. To estimate the error involved in the stress calculations, we define the percent relative error as jðrnumerical ranalytical Þ=ranalytical j 100. Fig. 7 shows that, as anticipated from the discussion above, the rxx
s xx
G 0.002
— 0.002
120
140
160
180
x [b] 200
— 0.004 — 0.006 — 0.008 — 0.01 — 0.012
Fig. 4. rxx =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude. The solid line is the analytical solution. The dashed lines are for the numerical solution. The larger the dash size, the denser the dislocation loop density (4, 5, 6, 8 10, 15, 20, 30, and 40 loops). Here, m ¼ 0:3 and r0 ¼ 100b. For Figs. 5–11, this Poisson ratio, particle radius and dashing system hold.
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
331
s yy G 0.025
0.02 0.015 0.01 0.005 120
140
160
180
x [b] 200
Fig. 5. ryy =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude.
s zz
G 0.002 0.00175 0.0015 0.00125 0.001 0.00075 0.0005 0.00025 120
140
160
180
x [b] 200
Fig. 6. rzz =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude. Here, the analytical solution is rzz ¼ 0.
s xx
% error
140 120 100 80 60 40 20 120
140
160
180
x [b] 200
Fig. 7. Percent relative error in rxx for the curves in Fig. 4.
error decreases with both increasing distance away from the particle (for any given loop density), and with increased loop density. The error plots for ryy and rzz are shown in Figs. 8 and 9 and reveal similar behavior.
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J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334 s yy % error
25 20 15 10 5
120
140
160
180
x [b] 200
Fig. 8. Percent relative error in ryy for the curves in Fig. 5.
s zz % error
30 25 20 15 10 5 120
140
160
180
x [b] 200
Fig. 9. Percent relative error in rzz for the curves in Fig. 6.
ux b — 0.1
200
300
400
500
x [b] 600
— 0.2 — 0.3 — 0.4 — 0.5 — 0.6 — 0.7 — 0.8
Fig. 10. ux =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude.
So far the results have concentrated on stress calculations. The displacement calculations are now presented. Fig. 10 shows the numerical solution approaching or converging towards the analytical one as the distance from the particle increases or the dislocation loop density increases. This is shown more clearly in Fig. 11 where the error decreases with distance from the particle and with increased loop density. Notice in these last two figures that ur ¼ ux along the x-axis.
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333
ux % error 50 40 30 20 10 200
300
400
500
x [b] 600
Fig. 11. Percent relative error in ux for the curves in Fig. 10.
s
% Error s xx
30 25
s yy
20 15
s zz
10 5 10
20
30
40
Ne
Fig. 12. A plot of the error in estimating stresses (rxx , ryy , and rzz ), at a point away from the particle a distance equal to the side of an approximating polygon, versus the number of sides. The number of polygon sides (or dislocation loops) is indicated by Ne.
Fig. 12 shows the error in stress calculation versus the number of dislocation loops, where stress is evaluated at a distance from the particle, d, equal to the length of an approximating polygon side. This figure facilitates a fast rule of thumb: using any polygonal approximation greater or equal to 10 dislocation loops, the stress calculation will be less than 10% in error if stress is evaluated at d or greater away from the particle.
4. Conclusions In this paper, a numerical distributed-dislocation method has been presented to solve generalized eigenstrain problems in 2D and was particularly applied to volume misfit particles. The method replaces the continuous misfit region at the particle-matrix interface with a continuous set of local misfit regions each of which is a prismatic dislocation loop. The methodÕs accuracy increases with dislocation loop density or number of loops. As a fast rule of thumb, there is a 10% or less error from analytical solutions if one calculates stresses away from the particle a distance equal to the size of a dislocation loop or larger.
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J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
Acknowledgements This work was supported (in part) by Lawrence Livermore National Laboratory (LLNL) under contract number B519473. In particular, the authors would like to thank the ‘‘LLNL Research Collaboration Program for HBCUs and other MIs’’ for its support. In addition, this work was supported (in part) by the Air Force Office of Scientific Research, USAF, under award number F49620-01-1-0565. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.
References Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. A 241, 376– 396. Hirth, J.P., Lothe, J., 1982. Theory of Dislocations. Krieger Publishing Company, Malabar, FL. Hull, D., Bacon, D.J., 1984. Introduction to Dislocations. Butterworth-Heinemann. Kreyszig, E., 1993. Advanced Engineering Mathematics. John Wiley & Sons, New York. Mura, T., 1982. Micromechanics of Defects in Solids. Martinus Nijhoff Publishers, The Hague, The Netherlands. Teodosiu, C., 1982. Elastic Models of Crystal Defects. Springer-Verlag, Berlin. Weertman, J., 1996. Dislocation Based Fracture Mechanics. World Scientific, Singapore.
Mechanics Research Communications 30 (2003) 325–334 www.elsevier.com/locate/mechrescom
The elastic fields of misfit cylindrical particles: a dislocation-based numerical approach Jesus Lerma a, Tariq Khraishi b
a,*
, Yu-Lin Shen a, Brian D. Wirth
b
a Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA Materials Science and Technology Division, Lawrence Livermore National Laboratory, Livermore, California 94551, USA
Abstract Analytical solutions for the displacement and stress fields due to a misfit particle in a host matrix can be difficult to obtain, especially for complex particle geometries. In this work, we present a numerical method for finding such fields in the case of infinitely-long particles. The method is based on discretizing the continuous misfit region between the particle and matrix into local misfit regions consisting of interstitial dislocation loops. The results presented here indicate very good agreement with analytical solutions and better convergence with increasing loop density. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Dislocation mechanics; Eigenstrains; Particle strengthening
1. Introduction The problem is depicted in Fig. 1, which shows the cross-section of an infinitely-long misfit particle, of radius r0 , placed into a smaller size hole within an infinite matrix. The radial misfit between the particle and the hole is denoted as d. In this figure, the particle and hole are shown to be perfect cylinders with uniform d along the interface. In general this need not be the case and is only shown here for ease of illustration of the present method. Also, the elastic constants of the particle can be different from those of the matrix. In micromechanics, it is desirable to quantify the displacement and stress fields, in the particle and matrix materials, caused by the presence of such misfit particles. Ideally, the solution is sought for any particle geometry or shape and any distribution of the misfit along the interface. Analytical solutions are typically derived for simplified particle geometries as well as for well-described or behaved misfit distributions. For example, Teodosiu (1982) solved the boundary-value problem (BVP) of a spherical misfit particle of uniform radial misfit and of different elastic properties than those of the matrix. Below we concisely follow TeodosiuÕs approach to give the solution in the case of a cylindrical particle, i.e. the 2D problem. This solution is provided for completeness and to compare with the numerical method presented in this paper. The main disadvantage of the BVP method is that every time the geometry of the problem changes, the
*
Corresponding author. Tel.: +1-505-277-6803; fax: +1-505-277-1571. E-mail address: [email protected] (T. Khraishi).
0093-6413/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00035-1
326
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
δ
r0
R
G',v' G,v
Fig. 1. A cylindrical particle of radius r0 inserted into a smaller hole in a cylindrical shell/matrix of radius R. The amount of misfit between the particle and the matrix is denoted by d. The particle-matrix interface can be approximated with a polygonal prism like the the hexagon shown in the figure.
solution must be recreated. Of course, the analytical solution for an infinite cylindrical particle with constant or uniform misfit can also be deduced as a special case from more advanced analytical solutions. For example, it can be deduced from the solution for an ellipsoidal particle presented by Eshelby (1957), among others, using the eigenstrain method. Mura (1982) presents an excellent treatise of the eigenstrain method for solving particle problems, including misfit situations. However, once again, every time the problem geometry changes, the solution must be re-formulated, and the eigenstrains are typically restricted to simple analytical functions. Below, we present a distributed-dislocation method for numerically solving particle misfit problems of any geometry and any misfit distribution along the interface. Here the state of continuous misfit region along the particle-matrix interface is replaced by a distribution of local misfits. Each local misfit region is represented by a prismatic interstitial dislocation loop. As the dislocation loop density increases, i.e. the number of loops increase and the size of each decrease, the numerical solution converges to the analytical solution. Here the Burgers vector of each loop corresponds to the local amount of misfit between the particle and the matrix. Due to its numerical nature, the method can be used to quantify the elastic fields of a particle of any geometry and any misfit distribution.
2. Method Fig. 1 shows the cross-section of an infinitely-long elastic cylindrical particle inserted into a smaller cylindrical hole of an elastic shell. The shell radius is denoted by R. The hole radius is smaller than the particle radius r0 by a radial misfit d (or equivalently by a misfit volume dt0 ¼ dð2pr0 Þ per unit length along the particle). In order to analytically solve for the elastic fields induced by this misfit, one has to solve an elasticity BVP subject to certain boundary conditions. Due to the cylindrical symmetry of the deformation field with only a radial component of displacement of relevance, we here use a semi-inverse method starting with a functional form for the displacement: ur ¼ u ¼ uðrÞ;
uh ¼ uz ¼ 0
ð1Þ
Notice that the displacement is independent of the angular position h of a material point P located at ðr; hÞ or equivalently at ðx; yÞ. Substituting Eq. (1) into the typical definition of strain in cylindrical coordinates, one obtains: err ¼
ou ; or
u ehh ¼ ; r
ezz ¼ erh ¼ ehz ¼ ezr ¼ 0
ð2Þ
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327
Assuming linear elasticity and isotropic particle and shell/matrix materials, HookeÕs law applies and one can find the stresses directly from Eq. (2), i.e. in terms of displacement. The stresses, however, must satisfy the static equations of equilibrium. In the absence of body forces, two of three equilibrium equations are identically satisfied or null. The third, after some algebraic simplifications, can be expressed as: drrr 1 þ ðrrr rhh Þ ¼ 0 r dr
ð3Þ
Substituting the stresses, in terms of displacement, into Eq. (3), one obtains: d2 u 1 du 1 u¼0 þ dr2 r dr r2
ð4Þ
Eq. (4) is a standard Cauchy–Euler differential equation (Kreyszig, 1993) and its solution has the general form: u ¼ uðrÞ ¼ C1 r þ C2 r1
ð5Þ
where C1 and C2 are integration constants. It is to be noted that all of the above equations apply to both the particle and the surrounding matrix. In order to distinguish between the two solutions we will hereafter add a prime to the integration constants and other field quantities belonging to the particle. The solutions for the particle and the matrix are then represented by Eqs. (6) and (7), respectively. u0 ¼ u0 ðrÞ ¼ C10 r þ C20 r1
ð6Þ
u ¼ uðrÞ ¼ C1 r þ C2 r1
ð7Þ
C20
must be equal to zero otherwise the displacement at the center of the particle In Eq. (6), the constant would be infinite. Eq. (6) is then reduced to: u0 ¼ u0 ðrÞ ¼ C10 r
ð8Þ
To solve for C1 , C2 , and C10 we need to apply boundary conditions (BCs). The pertinent BCs are: rrr ðr ¼ RÞ ¼ rrh ðr ¼ RÞ ¼ rrz ðr ¼ RÞ ¼ 0
ð9aÞ
rrr ðr ¼ r0 Þ ¼ r0rr ðr ¼ r0 Þ
ð9bÞ
2pr0 ½uðr ¼ r0 Þ u0 ðr ¼ r0 Þ ¼ dt0
ð9cÞ
With the aid of Eqs. (7–9), (2), and HookeÕs law we solve for the three constants (details skipped): C1 ¼
G C2 þ GÞ
R2 ðk
2 1 dt0 G r02 G r0 1þ 1 k þ G R2 k0 þ G0 R2 2p G 1 1 C10 ¼ 0 C2 k þ G0 R2 r02 C2 ¼
ð10aÞ
ð10bÞ
ð10cÞ
Eq. (10) and above can be used to explicitly solve for the displacement, strain, and stress fields. Note that in Eq. (10), G and G0 stand for the shear modulus of the matrix and particle, respectively; k and k0 stand for LameÕs constant (note that k and G are also both called LameÕs constants). The expressions of Eq. (10) are for the case of a cylindrical particle inside a cylindrical shell of finite radius, where the material properties of the shell and the particle are distinct. In our numerical illustrations below, we specifically deal with the case
328
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
where the media surrounding the particle is infinite, i.e. R equals infinity, and where the particle and the surrounding media possess equivalent material properties. Modifying Eq. (10) for this particular problem we obtain: C1 ¼ 0
ð11aÞ
1 dt0 G 1þ 0 C2 ¼ 2p k þ G0
ð11bÞ
G C2 k0 þ G0 r02
ð11cÞ
C10 ¼
Eq. (11) can be utilized to solve for stresses, strains, and displacements. To illustrate the present numerical method, we consider for simplicity the case where G ¼ G0 , k ¼ k0 , and thus m ¼ m0 (where m is PoissonÕs ratio). The misfit region between the particle and the matrix is approximated with an enclosed polygon (e.g. see Fig. 1). Each side of the polygon stands for a local misfit region. This local misfit representation is facilitated by considering the face of each side to be a prismatic dislocation loop (i.e., one whose Burgers vector b is pointed normal to its habit plane). The dislocation loop in this case is rectangular and is composed of four edge dislocation segments (two of them at z 1 Fig. 2). A prismatic dislocation loop is simply a layer of matter inserted or removed in a surrounding matrix (Hirth and Lothe, 1982; Hull and Bacon, 1984). Fig. 2 shows an oblique view of a section of the enclosed polygon. Each side has a Burgers vector normal to its face. The dislocation segments at the end of the infinite prism do not contribute to the state of elastic deformation in the material. Fig. 3 shows a cross-section of the infinitely-long prism (normal to its axis), with each prismatic dislocation loop being represented by two oppositely-signed, infinite edge dislocations. The elastic fields (displacement, strain and stress) at a field point P ðx; yÞ are numerically found to be, from the principle of linear superposition, the summation of the elastic fields due to the edge dislocations in Fig. 3. Note here that the Burgers vector of each dislocation i, bi , can be decomposed into two components: a glide component bix , and a climb component biy (the terminology of glide and climb components has been
b =δ Fig. 2. A section of the prism in Fig. 1. Each of the surfaces or sides stand for a prismatic dislocation loop, i.e. whose Burgers vector is normal to its plane. The magnitude of the Burgers vector, b, is equal to the misfit between the particle and the matrix d.
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
329
p Edge Dislocation
y
Dislocation Line Sense
Fig. 3. A cross-section of Fig. 2. Each dislocation loop is represented by two oppositely-signed, infinite edge dislocations. Point P is a material point that can reside inside or outside the particle. Also shown is a dislocation i whose Burgers vector is decomposed into x and y components.
borrowed from Weertman, 1996). As explained above, the displacements (ux and uy ) and stresses (rxx , rxy , and ryy ) fields at a material or field point will then be given by: ux ¼
2N X
uix ;
uy ¼
i¼1
rxx ¼
2N X
uiy
ð12Þ
i¼1
2N X
rixx ;
rxy ¼
i¼1
2N X i¼1
rixy ;
ryy ¼
2N X i¼1
riyy ;
rzz ¼
2N X
rizz
ð13Þ
i¼1
Notice that for N sides of the polygon, 2N edge dislocations exist (see Fig. 3) and hence the 2N summation limit in (12) and (13). The stresses and displacements at ðx; yÞ due to a single infinite edge dislocation i located at ðxi ; y i Þ with a Burgers vector bi ¼ bix þ biy are given by (Weertman, 1996): rixx ¼
2 2 2 2 Gbiy Gbix ðy y i Þð3ðx xi Þ þ ðy y i Þ Þ ðx xi Þððx xi Þ ðy y i Þ Þ þ 2 2 2 2 2 2 2pð1 mÞ 2pð1 mÞ ððx xi Þ þ ðy y i Þ Þ ððx xi Þ þ ðy y i Þ Þ
ð14aÞ
riyy ¼
2 2 2 2 Gbiy Gbix ðy y i Þððx xi Þ ðy y i Þ Þ ðx xi Þð3ðy y i Þ þ ðx xi Þ Þ þ 2pð1 mÞ 2pð1 mÞ ððx xi Þ2 þ ðy y i Þ2 Þ2 ððx xi Þ2 þ ðy y i Þ2 Þ2
ð14bÞ
rixy ¼
2 2 2 2 Gbiy Gbix ðx xi Þððx xi Þ ðy y i Þ Þ ðy y i Þððx xi Þ ðy y i Þ Þ þ 2 2 2 2 2 2 2pð1 mÞ 2pð1 mÞ ððx xi Þ þ ðy y i Þ Þ ððx xi Þ þ ðy y i Þ Þ
ð14cÞ
rizz ¼ mðrixx þ riyy Þ
ð14dÞ
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J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
# bix y yi 1 ðx xi Þðy y i Þ 1 þ tan ¼ 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 2p x xi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 i 2 ðx xi Þ2 þ ðy y i Þ2 biy ð1 2mÞ 1 ðy y Þ 5 ln þ þ 4 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 rc 2p 2ð1 mÞ "
uix
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 i Þ2 þ ðy y i Þ2 i i 2 ðx x b 1 2m 1 ðy y Þ 5 uiy ¼ x 4 ln þ 2ð1 mÞ 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 rc 2p " # biy y yi 1 ðx xi Þðy y i Þ 1 : tan þ 2ð1 mÞ ðx xi Þ2 þ ðy y i Þ2 2p x xi
ð15aÞ
ð15bÞ
In the above equations, rc is a cut-off or dislocation core radius taken here equal to the magnitude of one Burgers vector, b, in our numerical examples presented below.
3. Results and discussion We now illustrate the results of the above method. In Fig. 4, rxx is plotted along the x-axis for m ¼ 0:3 and r0 ¼ 100b. Note that along the x-axis, rrr corresponds to rxx , and rhh corresponds to ryy . The figure shows different curves each corresponding to a given dislocation loop density (the loop densities used are 4, 5, 6, 8 10, 15, 20, 30, and 40 loops, represented by the increasing dash size in Figs. 4–11). As the loop density increases, so does the dash size in the curves. We note in Fig. 4 that, for any given loop density, as the distance from the particle increases the numerical solution approaches the analytical solution (in solid line). For points relatively close to the particle (i.e. at x ¼ 100b), the numerical solution is only accurate for relatively large dislocation densities. Overall, the numerical solution converges to the analytical one as the loop (or mesh) density increases. Figs. 5 and 6 show similar trends, as above, for the cases ofryy and rzz , respectively. To estimate the error involved in the stress calculations, we define the percent relative error as jðrnumerical ranalytical Þ=ranalytical j 100. Fig. 7 shows that, as anticipated from the discussion above, the rxx
s xx
G 0.002
— 0.002
120
140
160
180
x [b] 200
— 0.004 — 0.006 — 0.008 — 0.01 — 0.012
Fig. 4. rxx =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude. The solid line is the analytical solution. The dashed lines are for the numerical solution. The larger the dash size, the denser the dislocation loop density (4, 5, 6, 8 10, 15, 20, 30, and 40 loops). Here, m ¼ 0:3 and r0 ¼ 100b. For Figs. 5–11, this Poisson ratio, particle radius and dashing system hold.
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334
331
s yy G 0.025
0.02 0.015 0.01 0.005 120
140
160
180
x [b] 200
Fig. 5. ryy =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude.
s zz
G 0.002 0.00175 0.0015 0.00125 0.001 0.00075 0.0005 0.00025 120
140
160
180
x [b] 200
Fig. 6. rzz =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude. Here, the analytical solution is rzz ¼ 0.
s xx
% error
140 120 100 80 60 40 20 120
140
160
180
x [b] 200
Fig. 7. Percent relative error in rxx for the curves in Fig. 4.
error decreases with both increasing distance away from the particle (for any given loop density), and with increased loop density. The error plots for ryy and rzz are shown in Figs. 8 and 9 and reveal similar behavior.
332
J. Lerma et al. / Mechanics Research Communications 30 (2003) 325–334 s yy % error
25 20 15 10 5
120
140
160
180
x [b] 200
Fig. 8. Percent relative error in ryy for the curves in Fig. 5.
s zz % error
30 25 20 15 10 5 120
140
160
180
x [b] 200
Fig. 9. Percent relative error in rzz for the curves in Fig. 6.
ux b — 0.1
200
300
400
500
x [b] 600
— 0.2 — 0.3 — 0.4 — 0.5 — 0.6 — 0.7 — 0.8
Fig. 10. ux =G vs x along the x-axis (for y ¼ 0). x is in units of b, Burgers vector magnitude.
So far the results have concentrated on stress calculations. The displacement calculations are now presented. Fig. 10 shows the numerical solution approaching or converging towards the analytical one as the distance from the particle increases or the dislocation loop density increases. This is shown more clearly in Fig. 11 where the error decreases with distance from the particle and with increased loop density. Notice in these last two figures that ur ¼ ux along the x-axis.
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333
ux % error 50 40 30 20 10 200
300
400
500
x [b] 600
Fig. 11. Percent relative error in ux for the curves in Fig. 10.
s
% Error s xx
30 25
s yy
20 15
s zz
10 5 10
20
30
40
Ne
Fig. 12. A plot of the error in estimating stresses (rxx , ryy , and rzz ), at a point away from the particle a distance equal to the side of an approximating polygon, versus the number of sides. The number of polygon sides (or dislocation loops) is indicated by Ne.
Fig. 12 shows the error in stress calculation versus the number of dislocation loops, where stress is evaluated at a distance from the particle, d, equal to the length of an approximating polygon side. This figure facilitates a fast rule of thumb: using any polygonal approximation greater or equal to 10 dislocation loops, the stress calculation will be less than 10% in error if stress is evaluated at d or greater away from the particle.
4. Conclusions In this paper, a numerical distributed-dislocation method has been presented to solve generalized eigenstrain problems in 2D and was particularly applied to volume misfit particles. The method replaces the continuous misfit region at the particle-matrix interface with a continuous set of local misfit regions each of which is a prismatic dislocation loop. The methodÕs accuracy increases with dislocation loop density or number of loops. As a fast rule of thumb, there is a 10% or less error from analytical solutions if one calculates stresses away from the particle a distance equal to the size of a dislocation loop or larger.
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Acknowledgements This work was supported (in part) by Lawrence Livermore National Laboratory (LLNL) under contract number B519473. In particular, the authors would like to thank the ‘‘LLNL Research Collaboration Program for HBCUs and other MIs’’ for its support. In addition, this work was supported (in part) by the Air Force Office of Scientific Research, USAF, under award number F49620-01-1-0565. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.
References Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. A 241, 376– 396. Hirth, J.P., Lothe, J., 1982. Theory of Dislocations. Krieger Publishing Company, Malabar, FL. Hull, D., Bacon, D.J., 1984. Introduction to Dislocations. Butterworth-Heinemann. Kreyszig, E., 1993. Advanced Engineering Mathematics. John Wiley & Sons, New York. Mura, T., 1982. Micromechanics of Defects in Solids. Martinus Nijhoff Publishers, The Hague, The Netherlands. Teodosiu, C., 1982. Elastic Models of Crystal Defects. Springer-Verlag, Berlin. Weertman, J., 1996. Dislocation Based Fracture Mechanics. World Scientific, Singapore.