By SREEKANTH AKARAPU - Washington State By Sreekanth Akarapu, Ph.D. Washington State University August

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  • DISLOCATION INTERATIONS WITH INTERFACES

    By

    SREEKANTH AKARAPU

    A dissertation submitted in partial fulfillment of

    the requirements for the degree of

    DOCTOR OF PHILOSOPHY

    WASHINGTON STATE UNIVERSITY

    School of Mechanical and Materials engineering

    AUGUST 2009

  • ii

    To the Faculty of Washington State University:

    The members of the Committee appointed to examine the dissertation of SREEKANTH

    AKARAPU find it satisfactory and recommend that it be accepted.

    ___________________________________

    Hussein Zbib, Ph.D. Chair

    ___________________________________

    Sinisa Mesarovc, Ph.D.

    ___________________________________

    David Field, Ph.D.

    ___________________________________

    Alexander Panchenko, Ph.D.

  • iii

    ACKNOWLEDGEMENTS

  • iv

    DISLOCATION INTERACTIONS WITH INTERFACES

    ABSTRACT

    By Sreekanth Akarapu, Ph.D.

    Washington State University

    August 2009

    Chair: Hussein M Zbib

    In this dissertation work, our main focus was to investigate the interactions of dislocation

    with interfaces. Plastic deformation in polycrystalline materials and multi-layered metallic

    composites, on a microscopic scale, involve interaction of dislocations with grain boundaries and

    bi-material interfaces respectively. Towards the end of investigating the interaction of

    dislocations with bi-material interface, we have derived analytical expressions for the stress field

    due to an arbitrary dislocation segment in an isotropic inhomogeneous medium. We have

    developed a new approach as compared with attempts made in the literature. One of the main

    advantages our derivation is separation of solution into homogeneous and image parts which

    facilitates an easy modification of existing dislocation dynamics simulation codes to incorporate

    the image stress effect.

    In the case of polycrystalline materials, as grain boundaries are major obstacles to plastic

    deformation, it is of fundamental importance to study the interactions of dislocations with grain

    boundaries. Towards this goal, in chapter four, we have investigated the basic phenomena of

    transmission of dislocation through a pure tilt wall. In this work, we have studied the structure of

    the symmetric tilt wall acquired after transmission of several dislocations and modeled the

    structures to which it relaxes.

  • v

    In chapter five, digressing from the main theme of the dissertation, we have studied the

    kinematic and thermodynamics effect of representing discrete dislocations in terms of

    continuously distributed dislocations. In this work, we have considered infinite stacked double

    ended pile-ups in an isotropic elastic homogeneous medium. The error in number of dislocations,

    microstructural energy and slip distribution between discrete and semi-discrete representation

    was quantified. The asymptotic expressions are derived and threshold values of certain key

    parameters are also deduced.

    In the appendix, we have investigated the deformation of single crystal micropillars under

    uniaxial compression using a multi-scale model for plasticity. Our simulation results are

    qualitatively and quantitatively comparable with that of experiments. Dislocation arm operation

    was found to be the prominent mechanism to plastic deformation in micron to submicron size

    specimens. The observed strain hardening is attributed to the formation of entangled dislocation

    structures and stagnation of dislocations.

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    TABLE OF CONTENTS

    Acknowledgements ........................................................................................................................ iii

    Abstract .......................................................................................................................................... iv

    List of Figures ................................................................................................................................ ix

    List of tables ................................................................................................................................. xiii

    Chapter One: Introduction .............................................................................................................. 1

    Chapter Two: A Unified approach to dislocation stress fields in dislocation dynamics simulations

    ......................................................................................................................................................... 9

    2.1 Introduction ........................................................................................................................... 9

    2.2.1 Anisotropic Greens functions derivatives .................................................................... 17

    2.2.2 Mura‟s Integral with Anisotropic Greens Tensor Derivatives ..................................... 20

    2.3 Continuous distribution of dislocations/ Long range interactions ...................................... 24

    2.4 Summary ............................................................................................................................. 26

    Chapter Three: Line-Integral Solution for the Stress and Displacement Fields of an Arbitrary

    Dislocation Segment in Isotropic Bi-materials in 3D Space ........................................................ 27

    3.1 Introduction ......................................................................................................................... 29

    3.2. Methodology ...................................................................................................................... 33

    3.2.1) Bonded Interface......................................................................................................... 34

    3.2.2) Dislocation segment ................................................................................................... 41

    3.3. Infinite edge dislocation ..................................................................................................... 42

    3.3.1) Isotropic joined half space with bonded interface ...................................................... 42

    3.3.2) Isotropic half space with traction-free boundary ........................................................ 47

    3.3.3) Isotropic half space with rigid boundary .................................................................... 48

    3.3.4) Interface dislocation ................................................................................................... 48

    3.3.5) Circular dislocation loop ............................................................................................ 48

    3.4. Conclusions ........................................................................................................................ 49

    Appendix 3.A ............................................................................................................................ 51

    Chapter Four: Dislocation Interactions with Tilt Walls ................................................................ 61

    4.1 Introduction ......................................................................................................................... 61

    4.2 Effect of Pile-up dislocations .............................................................................................. 63

    4.3 Dislocation transmission ..................................................................................................... 64

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    4.4. Stress computation method ................................................................................................ 66

    4.5. Disconnections and disclination dipoles ............................................................................ 68

    4.6 Summary ............................................................................................................................. 71

    Chapter Five: Energies and distributions of dislocations in stacked pile-ups .............................. 82

    5.1 Introduction ......................................................................................................................... 84

    5.2 Representations of geometrically necessary dislocations and the microstructural energy . 86

    5.3 Formulation of the problem and numerical methods .......................................................... 88

    5.3.1 Semi-discrete representation ........................................................................................ 88

    5.3.2 Asymptotic solutions for the semi-discrete representation .......................................... 92

    5.3.3 Discrete representation................................................................................................. 96

    5.4 Numerical results and analysis............................................................................................ 98

    5.4.1 Number of dislocations in a pile-up and the microstructural energy ........................... 99

    5.4.2 Slip distributions ..........................................................................................