Prediction of stress-strain curves and study of incompatibility stresses using Crystal Plasticity Finite Element Method (CPFEM)

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Frank-Read source

Among many different dislocation source models, such as Frank-Read source, condensation of point defects, particles and inclusions, grain boundary sources, surface sources and crack sources, etc. [41], the Frank-Read source is an im-portant mechanism of dislocation multiplication and has been largely studied [42]. This mechanism involves the curvature of a dislocation wedged between two points under the effect of its line tension, and the subsequent instability of the curved dislocation which leads to the formation of a dislocation loop and the maintenance of the original trapped dislocation (see Figure 1.9). This source was observed ex-perimentally by Dash in a silicon sample with Copper (Cu) precipitates [43]. This phenomenon has also been observed in other materials, particularly in Aluminum (Al) (see Figure 1.10) [44].

Stacking fault energy

As discussed in subsection 1.1.1.2, a perfect dislocation can be separated into several partial dislocations when certain conditions are met. However, the energy before decomposition must be greater than the sum of the energy of each part after decomposition, i.e. the process should be thermodynamically favorable. When a perfect dislocation is decomposed on a certain crystal plane, a stacking fault is formed. The perfect lattice structure between the dislocations of the various faces on the crystal face is further broken, which causes the rise of systematic energy. This energy is called stacking fault energy. Only when the metal stacking fault energy is low, the decomposition of complete dislocations will appear.
In a crystal, when two or more different slip planes slide simultaneously or alternately along a common slip direction, the slip is called cross-slip. Only screw dislocations can be transferred from one slip plane to another one. The stacking fault energy has a significant influence on the formation of dislocations and the cross slip phenomenon. When the stacking fault energy is low, dislocations are easily decomposed into partial dislocations and cross-slip cannot occur, thus planar slip is promoted. However, when the stacking fault energy is high, cross-slip is promoted and it will cause more homogeneous slip.

GB geometry and bi-crystallography

The classical GB is regarded as a plane interface defined by a normal vector between two crystals of different orientations, while the atoms distortion close to GB is disregarded. This GB is just an infinitely thin geometrical boundary separating two grains without any specific properties.
A complete geometrical characterization of GB can be specified by 9 geometri-cal degrees of freedom which include five macroscopic degrees of freedom and four microscopic degrees of freedom [50, 51]. The macroscopic degrees of freedom in-clude misorientation and interfacial plane which are required to define a bi-crystal from given crystals:
Three degrees of freedom are necessary to describe the misorientation be-tween grains: one to define the rotation angle θ, two to define the rotation axis r = huvwi. The misorientation angle is the smallest rotation angle to get from crystal to another.
Two degrees of freedom are necessary to describe the inclination of the GB plane defined by its unit normal n.
The microscopic degrees of freedom are needed to describe the atomic structure of GB which are determined by relaxation processes:
Three degrees of freedom are necessary to describe the translation of one crystal to another one. The rigid body translation vector τ includes two translations in the GB plane and one for expansion perpendicular to the GB plane.
One degree of freedom is necessary to describe the position d of the GB plane along its normal. The value of d should be less than 1.
Depending on the misorientation angle θ, GBs can be separated into low angle GBs (LAGB) with θ < 10◦ and high angle GBs (HAGB) with θ > 15◦.

Twist and tilt grain boundaries

The GB can be firstly categorized by the relationship between the rotation axis r to get from grain to another and the GB normal vector n. If the rotation axis is parallel to GB normal (r k n), the GB is called a twist GB as shown in Figure 1.15 (a). In contrast, if the rotation axis is perpendicular to the GB normal (r ⊥ n), the type GB is called a tilt GB as shown in Figure 1.15 (b). A real GB is generally neither pure twist nor pure tilt, but combines the two types together. This last case is called a fixed GB as shown in Figure 1.15 (c).

Coherent and incoherent grain boundaries

Considering the atom matching at GB, GBs can also be classified into another three types: coherent GB, semicoherent GB and incoherent GB [51]. It should be pointed out that this classification is generally used for interphase boundaries, which are boundaries between two different phases. Two different phases may have different compositions, crystal structure and/or lattice parameter.
Definition of coherent GB: Both crystals match perfectly at the interface plane as shown in Figure 1.16 (a). The interfacial plane has the same atomic configuration in both crystals like twin boundaries as shown in Figure 1.17. The energy of a perfectly coherent interface is very low: on the order of a few mJ · m−2. Small lattice mismatch at GB can be accommodated by elastic strain and coherent interface can be maintained as shown in Figure 1.16 (b).

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Continuum based GB dislocation model

A GB can be described as a GB dislocation characterized by a Burgers vector density through the continuous Frank-Bilby (FB) approach [51]. This Burgers vector density bp which is necessary to realize the compatibility at GB between two crystals I and II can be calculated by the Frank-Bilby equation [53, 54]: bp = SI−1 − SII−1 · p (1.14).
where p is a periodic vector in GB plane, SI and SII are the transform matrices that generate the lattice of crystal I and crystal II from a reference lattices as shown in Figure 1.21. bp is the Burgers vector content of all the dislocations that are crossed by the vector p. The FB equation can be also used for HAGB even though discrete dislocation model cannot be defined anymore due to dislocation overlap.

Atomic Force Microscopy

Atomic Force Microscopy (AFM) is an atomic-level high-resolution analytical instrument invented by G. Binning in 1986 based on Scanning Tunneling Micro-scope (STM). It can perform nano-scale studies of physical properties including topographies in various materials. Compared with STM, AFM has a wider appli-cability because it can observe non-conductive samples. It has been widely used in the fields of semiconductor, nano-functional materials, biological, chemical, med-ical research and research institutes in various nano-related disciplines, and has become a basic tool for nanoscience research. Especially, slip lines or slip bands heights can be observed and analyzed by this technique [33, 34, 8, 35, 36]. Fur-thermore, based on the analyses of slip bands, the dislocation distribution of a pile-up against GB can be reformed considering a continuous distribution [125] or a discrete one in anisotropic elasticity as it will be discussed in the present PhD thesis.
The principle of AFM is described below. The one end of a weakly sensitive cantilever is fixed, with another side as a microprobe which is in light contact with the sample surface. Due to the distance-dependent weak repulsion between the atoms on the probe tip and those on the sample surface, by keeping this force constant (which corresponds to a constant distance) during scanning, the probe’s undulating motion perpendicular to the sample surface reflects the height of sample surface. The topographic information of sample surface is thus obtained based on this surface height profile. The tip motion corresponding to each scanning point can be measured by optical detection as shown in Figure 1.41.

Table of contents :

1 State of the art 
1.1 Dislocation and crystal plasticity
1.1.1 Introduction to dislocations
1.1.2 Slip mechanism
1.1.3 Deformation Twinning
1.2 Grain boundary and free surface
1.2.1 Denition of a grain boundary
1.2.2 Denition and classication of grain boundaries
1.2.3 Free surface eects
1.3 Interaction between dislocation and grain boundary
1.3.1 Experimental observations
1.3.2 Dislocation pile-up
1.3.3 Image force and image dislocation
1.3.4 Slip transmission and geometrical criteria
1.3.5 Grain boundary strength
1.4 Experimental methods
1.4.1 SEM and EBSD
1.4.2 Ion Milling
1.4.3 Focused Ion Beam technique
1.4.4 Atomic Force Microscopy
1.4.5 Micro-beam size eects
1.5 Theoretical and numerical multi-scale modelling methods
1.5.1 Continuum dislocation mechanics
1.5.2 3D incompatibility stresses in bi-crystals
1.5.3 Two dimensional L-E-S (Leknitskii-Eshelby-Stroh) formalism for anisotropic elasticity
1.5.4 Crystal Plasticity Finite Element Method (CPFEM)
1.5.5 Discrete Dislocation Dynamics method (DDD)
1.5.6 Molecular Dynamics simulation (MD)
2 Experimental part: Nickel and -Brass bicrystalline micro-pillar compression test 
2.1 Introduction
2.2 Material choice
2.3 Sample preparation
2.3.1 Metallographic preparation
2.3.2 Heat treatment
2.3.3 Grain boundary choice
2.3.4 Micro-beam preparation
2.4 Preanalyses of slip information
2.5 Micro-pillar compression tests
2.6 Stress-strain analysis
2.7 Slip analysis by SEM and AFM
3 Elastic elds due to single dislocations and dislocation pile-ups in heterogeneous and anisotropic media 
3.1 Introduction
3.2 Elastic elds due to one single dislocation in dierent congurations
3.2.1 Homogeneous anisotropic medium
3.2.2 Heterogeneous anisotropic medium: bi-material
3.2.3 Anisotropic half-space with rigid or free surface
3.2.4 Heterogeneous anisotropic medium: tri-material
3.2.5 Heterogeneous anisotropic medium: multilayer material with free surfaces
3.3 Discrete dislocation pile-ups theory
3.4 Computational procedure
4 Results and discussions 
4.1 Introduction
4.2 Theoretical results
4.2.1 Computation congurations
4.2.2 Convergence of the series solutions within the tri-material conguration
4.2.3 Displacements and stresses distribution due to one single dislocation
4.2.4 Image force on dislocation in heterogeneous media
4.2.5 Results for discrete dislocation pile-ups
4.3 Prediction of stress-strain curves and study of incompatibility stresses using Crystal Plasticity Finite Element Method (CPFEM)
4.3.1 CPFEM conguration: geometry and mesh
4.3.2 Calibration of displacement and determination of material parameters
4.3.3 Incompatibility stresses
4.4 Computations of slip step height compared with experimental observations
4.4.1 Simulation conguration for experiment
4.4.2 Results of Ni micro-beam
4.4.3 Results of -Brass micro-beam
4.5 Conclusions of Chapter 4
Conclusions and Perspectives
Appendix
A Equivalence between the formulations of T.C.T. Ting and Z. Suo for the elastic elds of a dislocation in bi-materials with perfectly bonded interface
B Full coecient matrix of image decomposition method
C GB thickness and structure analyzed by Molecular Statics (MS) / Molecular Dynamics (MD) simulations
C.1 MD conguration
C.2 Results and discussions
Bibliographie 

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