3D DISCRETE DISLOCATION DYNAMICS APPLIED TO ... - NUMODIS

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20 Description of the simulation method Effective stresses After the internal (σint) and the applied stresses (σapp) are computed, the force on a slip system is defined by the Peach-Koehler equation and a projection along the glide direction g as shown in Eq.2.7. , where l is the unit vector tangent to the dislocation line. τg b = {[(σint + σapp) · b] × l} .g (2.7) It should be noted that σint and σapp are computed at the center of a given segment on the assumption that the stress field variations are small over the segment length. The effective stress τe is then computed by summing all the contributions as τe = τg + τlt − τp. Then, the velocity of the dislocation segment is given by Eq. 2.13. 2.2.2 Computation of displacements The computation of the displacement field of dislocations is very useful not only in analyzing surface deformation induced by dislocations, but also in imposing displacement boundary conditions in a coupling method with a finite element method (Sec. 2.4.2). The displacement solution of any closed curved dislocations can be found from the Burgers formula in the frame of elastic isotropy. The Burgers equation is given in terms of line and area integrals as shown in Eq. 2.8 in a vector form. u(r) = − b 1 b × dl Ω − 4π 4π C ′ R + 1 8π(1 − ν) ∇ (b × R) dl C ′ R b is the Burgers vector and ν is the Poisson ratio. Ω is the solid angle through which the positive side of a loop is seen and is defined as follows. RdA Ω = − A R3 The parameters for the computation Eq. 2.8 and Eq. 2.9 are shown for the configuration of a closed loop in Fig. 2.8. An analytical solution of the displacement fields can be obtained using Eq. 2.8 for the case of simple dislocation loops 7 . The solutions of complex dislocation loops are generally difficult to be resolved analytically. The general way of computing a displacement field of an arbitrary dislocation loop is to decompose the loop into triangular loops as illustrated in Fig. 2.8. The methodology to construct a displacement field from triangular loops was first presented by Hirth and Lothe (see 7 Khraishi et al. ([Khraishi et al. 00b]) have found a closed-form analytical solution of a circular dislocation loop. (2.8) (2.9)
2.2 Computation of stresses and displacements of dislocations 21 b C A Field point Ω R dl’ n Slip plane normal Dislocation loop b Triangular loop n Dislocation segments Figure 2.8: The parameters in the Burgers equation (Eq. 2.8) and decomposition of a dislocation loop by triangular dislocation loops [Hirth & Lothe 92]). Special care, however, should be taken at evaluating the inverse trigono- metric functions, as the author experienced. Barnett ([Barnett 85]) has developed a formula more suitable for numerical computation, which will be detailed below. The displacement at a field point P(r) generated by a triangular dislocation loop with points A(rA), B(rB) and C(rC) are expressed as Eq. 2.10. The triangular dislocation loop ABC and a field point are shown in Fig. 2.9. u(r) = − b 4π Ω + FAB + FBC + FCA (2.10) Ω is the solid angle associated with the triangle ABC, which generates a discontinuity of ∆u = b in traversing the cut surface ABC. F ij(i,j=A,B or C) is a continuous displacement field term ex- cept on the dislocation line. The solid angle Ω and the continuous terms Fij are given as follows ([Barnett 85]). s s − a s − b s − c Ω = −sign (Ri.n) 4 arctan tan tan tan tan 2 2 2 2 Fij = − 1 − 2ν 8π(1 − ν) (b × tij) ln Rj + Rj.tij Ri + Ri.tij + 1 8π(1 − ν) (b.nij) Rj Rj The vectors and the constants in Eq. 2.11 and Eq. 2.12 are listed below. − Ri × nij Ri (2.11) (2.12)
Page 1 and 2: INSTITUT NATIONAL POLYTECHNIQUE DE
Page 3: Acknowledgements First of all, I ex
Page 6 and 7: Résumé La dynamique des dislocati
Page 8 and 9: viii CONTENTS 2.3.5 Plastic strain
Page 11 and 12: Chapter 1 Introduction 1.1 Computat
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Page 15 and 16: 1.3 Scope of Thesis 5 Recently the
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Page 63 and 64: 3.1 An introduction to Supercomputi
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3.3 Parallelization of the serial D
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3.4 Performance improvment 81 Speed
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3.4 Performance improvment 87 Y Z I
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3.5 Application to Stage I-II trans
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Chapter 4 Dislocation-precipitate i
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4.1 Image stresses due to a 3D part
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4.2 A simple case of dislocation-pa
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Chapter 5 Conclusions and perspecti
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namics codes into a parallel versio
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Bibliography [Abraham 97] Abraham F
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BIBLIOGRAPHY 155 [Devincre & Robert
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BIBLIOGRAPHY 157 [Khraishi et al. 0
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BIBLIOGRAPHY 159 [Risbet et al. 03]
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