Cyclic Deformation in Single Crystals

Cyclic Deformation In Ductile Single Crystals

By Nandakishor V YaragattiRoll No: 100922006

Course: MTECH-CAMDACollege: MIT, Manipal

Contents• Formation of persistent slip bands• Electron microscopy observations• Static or energetic models• Dynamic models of self organized dislocation

structures• Formation of labyrinth and cell structures• Effects of crystal orientation and multiple slip• Monotonic V/s cyclic deformation in FCC Crystals• Cyclic deformation in BCC single crystals• Shape changes in fatigued BCC crystals• Cyclic deformation in HCP single crystals• Bibilography

Formation of PSBs• At the beginning of stage B in the saturation cyclic

stress-strain curve, structural changes must take place within the matrix to accommodate high values of plastic strains because the dislocation veins in the matrix cannot accommodate strains in excess of approximately 10-4.

Electron microscopy observations

• Hozwarth & Essman presented a study of the mechanism by which the matrix vein structure is transformed into the wall structure of a PSB.

• They started with a saturated matrix vein structure in a Cu single crystal at plastic shear strain, = 10-4, using a fully-reversed plastic-strain control test, until saturation to a cumulative plastic strain of =15 at shear stress, =28Mpa at 300K.

• The crystal was then subjected to a sudden increase in plastic shear strain, =4X10-4. which caused an instant jump in the flow stress to 33MPa and which initiated the formation of PSBs.

Formation of PSBs

• These experiments reveal that the transformation from the matrix vein structure to the PSB wall structure most likely commences at the centers of the veins wherein exist small areas that are dislocation poor.

• These soft areas are surrounded by a harder shell of higher dislocation density, wherein develop the first dislocation walls.

• In the plateau regime, the walls shift at a rate of 1-2nm/cycle, and this shift plays an important role in establishing the typical ladder pattern in PSB.

Evolution of PSB wall structure

• The evolution of a PSB wall structure in the dislocation-poor region of the matrix veins(marked by arrows). g=(1 1 1). =4X10-4 . (From Holzwarth & Essman,1993.)

Formation of PSBs

• Consider the PSB in the above fig. which cuts through a row of veins.

• From the geometry of the nascent wall structure and surrounding vein structure, it is noted that the walls originate from the vein shells and that they have to move very little to establish their spacing during PSB evolution.

Static or energetic models• The formation of dislocation structures within the veins

and PSBs can potentially be obtained from calculations of the equilibrium of positions finite population of dislocations.

• Basis for the earlier models was the Taylor-Nabarro lattice.

• According to Neumann, force per unit length on the i th dislocation is given by:

Static or energetic models

Static or energetic models

Dynamic Models of self-organized dislocation structures

• The dislocations are represented by density, which is a function of space and time.

• The to-and fro motion of dislocations under a cyclic stress is modeled in one dimension as a diffusion phenomenon with a flux term D, where D is an effective diffusion coefficient and rho is the density of dislocations.

• Two populations of dislocation are considered: immobile dislocation of density rho i and free dislocations of density rho m.


Walgraef and Aifantis arrived at the following set of coupled differential equations for densities of trapped and free dislocations:


• Solution of the above equation predicts the instabilities in the form of oscillations in time and spatial patterning.

• These two instabilities are related to the ocurrence of strain bursts and the formation of PSBs respectively.

• This analysis relies on the assumption of dislocations and ignores the specific dislocation geometries within dipolar PSB walls and channels.


• Differt and Essman proposed a dynamic model for edge dislocation walls within reaction transport modelling framework.

• Two important length scales were introduced in the reaction terms: (i) the critical annihilation distance of a dipole under the influence of an applied stress and (ii) the critical distance for the spontaneous annihilation of closely spaced dipoles.

• The analysis shows how the walls move.


• This mechanism rationalizes why a freshly formed PSB is less periodic and imperfect than a mature one.

• In summary, the dislocation arrangements in fatigue can be broadly classified in to two basic groups: (i) Structure in which equilibrium is maintained and (ii) Nonequilibrium self-organised dislocation structures.

• The nonequilibrium structures have been shown to provide a rationale for the instigation of fatigue instabilities such as the formation of ladder structures in PSBs and strain bursts.

Formation of labyrinth and cell structures



• For plastic shear strain <10-3 the slip band structure is characterized by the hard matrix comprising veins and the softer PSBs with wall structure.

• An increased contribution of secondary slip and a gradual evolution of ‘labyrinth’ and ‘cell’ structures are noticed for plastic shear strain > 2X 10-3.

• The labyrinth consists of two sets of orthogonal Burgers Vectors: b1 and b 2 denote the primary and conjugate Burgers Vectors respectively.


• At higher plastic shear strain and end of saturation stress strain curve for FCC crystals: matrix phase with labyrinth structure to PSBs and labyrinth structure to cell structure.

• Secondary slip (prevalent in region C) originates at the PSB-matrix interface and spreads in the form of an expanding cell structure which fills the PSBs.

• The transformation of all the PSBs into a cell structure appears to occur after 106. cycles.

Effects of crystal orientation and multiple slip

• The cyclic deformation of FCC single crystals oriented for single slip exhibits two prominent features ;(i) the existence of a plateau region, and (ii) the formation of PSBs with their characteristic wall structures.

• The possibility of occurrence of plateau in the CSS- curve, as well as the extent of the plateau region are strongly influenced by the crystallographic direction of the FCC crystal along which the fatigue loading is imposed


• Gong, Wang & Wang have studied the effect of multiple slip on CSS curve in Cu crystals. They found an absence of a plateau regime & in the strain range = 1x 10-4 – 3x10-3 , no PSBs were found to occur.

• Fatigued [001] crystals comprise principally labyrinth structures of primary and critical dislocations, which can more easily accommodate multiple slip and cross slip than the ladder structures.



• The interactions among different slip systems in the labyrinth and the attendant formation of lomer-cottrell locks causes a much higher cyclic hardening rate in multiple-slip orientations than in single glide.

• The labyrinths accommodate different imposed plastic strains by appropriately adjusting their channel widths; an increase on plastic shear strain is accomodated by a reduction in the channel width of the labyrinth.

• Multiple slip during cyclic deformation causes improvement in fatigue limit.






Monotonic v/s cyclic deformation in FCC crystals

• At very low plastic shear strains, (≤5X10-4), the dislocation configurations first generated during rapid cyclic hardening correlate well with the substructures found during stage I deformation of FCC monocrystals in monotonic tension, with the exception that the matrix veins seen in fatigue are akin to the cell structures in Stage II of monotonic tension.

• At higher strain amplitudes, the dislocation structures in the cyclic work hardening stage are similar to the unidirectional Stage I configurations only during the first few cycles.


• At high plastic strain amplitudes, corresponding to the regime of the CSS curve, the formation of cell structures and the associated rapid hardening during the early fatigue cycles finds an analogy in stage II deformation of FCC single crystals in unidirectional tension.

• The formation of PSBs at the onset of saturation is somewhat analogous to coarse slip band development during Stage III deformation of FCC single crystals in monotonic tension.


• The density of dislocations produced during cyclic loading is significantly higher than that generated, at comparable stresses, during monotonic tension.

• The evolution of persistent slip bands with their wall structure of edge dislocations is specific to cyclic deformation.

• A striking feature of fatigue deformation is the establishment of a saturated state where the peak resolved shear stress is independent of the plastic shear strain amplitude.


• The flow stress of FCC crystals exhibits a stronger dependence on temperature and strain rate in fatigue than in tension.

• Monotonic loading leads to the formation of surface slip steps which resemble a staircase geometry, cyclic deformation produces sharp peaks and valleys at sites where the PSBs emerge at the specimen surface.

• In monotonic tensile deformation of a single crystals, both the slip plane and the slip direction rotate toward the tensile axis, whereas there is no such orientation change during fully reversed cyclic loading of the crystals, however this results in prominence of primary dislocations and the absence of long-range internal stresses durin cyclic hardening.


Cyclic deformation in BCC single crystals

• The core of the screw dislocation in BCC metals doesn’t dissociate, and the particular nature of the screw dislocation core structure in BCC induces very high lattice friction.

• In BCC metals due to the special role of screw dislocations effects such as strain-rate sensitivity, strong temperature dependence of cyclic deformation, relative mobility of edge and screw dislocations, as well as asymmetry of slip between tension and compression are seen.


• Different regimes in the variation of mean saturation axial stress as a function of the axial plastic strain range during cyclic deformation of α-Fe single crystals at 295K.

• At low plastic strain amplitudes (≤10-3), essentially no hardening occurs and the cyclic strain is a manifestation of the motion of edge dislocations only.

• At higher strain amplitudes, deformation proceeds by the large scale motion of edge and screw dislocation and culminates in the formation of a cell structure; pronounced cyclic hardening as well as change in the shape of the crystal are observed due to asymmetric slip of screw dislocations in tension and compression.


• Although no PSBs have been identified in either regime of plastic strain amplitudes, ill defined bands of slip have been noticed which leads to the crack nucleation.

• In TEM investigations of dislocation structures ahead of fatigue cracks have identified the existence of PSBs in polycrystalline Cu but not in pure -Fe.

Difference in the fatigue response of BCC and FCC single crystals

• At 295K and at low plastic strain amplitudes, thermally activated glide of screw dislocations as well as dislocation multiplication are strongly suppressed in BCC α-Fe, whereas FCC metals are only weakly strain rate-sensitive, the flow stress of BCC metals is strongly dependent on the strain rate.

• In general, as a consequence of dynamic strain-ageing, high temperatures, very low strain rates and the addition of impurity atoms to the BCC metal promote cyclic damage that is similar to that found in FCC metals.

Shape changes in fatigued BCC crystals

• If slip occurs on different planes during tension and compression portions of fatigue, a crystal must undergo shape changes due to this slip symmetry.

• Neumann showed that the shape change produced by cyclic straining can be correlated with slip irreversibility, which is an important factor for crack nucleation.

• The net displacement after N fatigue cycles of a point in the crystals is given by the following relation:



Cyclic deformation in HCP single crystals

• The deformation is strongly influenced by impurity and interstitial content.

• Cyclic deformation is strongly influenced by the propensity for twin formation. An increase in the occurrence of cyclic twins causes a marked in the cyclic hardening rate.

• At fixed applied strain amplitudes, orientations which promote single slip and cross slip give rise to dipole arrays and dislocation loops, whereas cell structures are found in the specimens oriented for duplex and mulitple slip

Cyclic deformation in HCP single crystals

Bibliography • Suresh S• Thomas H Courtney• Effect of orientations on cyclic deformation

behavior of Ag and Cu single crystals: Cyclic stress–strain curve and slip morphology by P. Li, Z.F. Zhang *, S.X. Li, Z.G. Wang;Shenyang, National Laboratory for Materials Science, Institute of Metal Research.

• Science Direct, Acta materialia.

Cyclic Deformation in Single Crystals

Documents

dislocation veins

dislocation poor

dislocation arrangements

populations of dislocation

formation of ladder

edge dislocation walls

i th dislocation

psb wall structure