PLASTICITY

Dislocations and Materials Classes • Ionic Ceramics (NaCl): Motion hard. -need to avoid ++ and - - neighbors. + - • Covalent Ceramics (Si, diamond): Motion hard. -directional (angular) bonding • Metals: Disl. motion easier. -non-directional bonding -close-packed directions for slip. electron cloud ion cores +

Dislocation Motion Dislocations and plastic deformation Cubic & hexagonal metals - plastic deformation by plastic shear or slip where one plane of atoms slides over adjacent plane by defect motion (dislocations). So we saw that above the yield stress plastic deformation occurs. But how? In a perfect single crystal for this to occur every bond connecting tow planes would have to break at once! Large energy requirement Now rather than entire plane of bonds needing to be broken at once, only the bonds along dislocation line are broken at once. If dislocations don't move, deformation doesn't occur! Adapted from Fig. 7.1, Callister 7e.

Dislocation Motion Dislocation moves along slip plane in slip direction perpendicular to dislocation line Slip direction same direction as Burgers vector Edge dislocation Adapted from Fig. 7.2, Callister 7e. Screw dislocation

If a material is subjected to a load of sufficient magnitude it shows permanent (irrecoverable) deformation. It is result of the permanent displacement of atoms and molecules from their original position. If the deformation is continuously increasing then the phenemenon is called FLOW. P x’ x Pn Ps Slip direction Slip plane

Deformation Mechanisms Slip System: Slip plane and direction Slip plane - plane allowing easiest slippage - Highest planar densities Slip direction - direction of movement - Highest linear densities

FCC FCC Slip occurs on {111} planes (close-packed) in <110> directions (close-packed) => total of 12 slip systems in FCC

(111) (111) (111) (111) Parallel (111)

Therefore, for an FCC structure: So, for (111) plane: 3 (111), [101] or [101]---1 1 (111), [110] or [110]---2 2 (111), [011] or [011]---3 Therefore, for an FCC structure: {111} - <110> , there are 12 slip systems.

To sum up: Slip phenemenon is used to explain the plastic behaviour of materials. Slip occurs along certain crystal planes and directions. Slip planes & slip directions make slip systems. For an FCC structure {111} - <110> , there are 12 slip systems. For an BCC structure {110} - <111> → 12 slip systems. For an HCP structure → 3 slip systems.

The stress that initiates slip is known as the critical resolved shear stress. F Ø: Angle between the normal to the slip plane and the applied stress directions. λ: Angle between the slip and stress directions. A As τCR = σ . cos λ . cos Ø

Stress and Dislocation Motion • Crystals slip due to a resolved shear stress, tR. • Applied tension can produce such a stress. Applied tensile stress: = F/A s direction slip F A slip plane normal, ns Resolved shear stress: tR = F s /A direction slip AS FS direction slip Relation between s and tR = FS /AS F cos l A / f nS AS

Fs= F cos λ (shear force along the slip direction) As = A / cos Ø (shearing area) σ Fs F cos λ F τCR = = = cos λ . cos Ø A s A / cos Ø A τCR = σ cos λ cos Ø

Single Crystal Slip Adapted from Fig. 7.9, Callister 7e.

Ex: Deformation of single crystal a) Will the single crystal yield? b) If not, what stress is needed? =60° crss = 30 MPa =35° Adapted from Fig. 7.7, Callister 7e.  = 65 MPa So the applied stress of 65 MPa will not cause the crystal to yield.

Ex: Deformation of single crystal What stress is necessary (i.e., what is the yield stress, sy)? (!We will learn about yield stress later on!) So for deformation to occur the applied stress must be greater than or equal to the yield stress

Ex: Consider a single crystal of BCC iron oriented such that a tensile stress is applied along [010] direction. Compute the resolved shear stress along a (110) plane and in a [111] direction when a tensile stress of 52 MPa is applied. If slip occurs for the above plane and direction, and the critical resolved shear stress is 30 MPa, calculate the magnitude of the applied tensile stress necessary to initiate yielding.

Ø=45° angle b/w normal [110] & stress [010] [110] z Ø=45° angle b/w normal [110] & stress [010] [110] σ= 52 σ= 52MPa y (110) x cos Ø = 1*0 + 1*1 + 0*0 (12+12+02) (02+12+02) = 1 √2 Ø = 45° λ: angle b/w slip direction [111] & stress direction [010] cos λ = -1*0 + 1*1 + 1*0 (12+12+12) (1) = 1 √3 λ = 54.7° τCR = 52 * cos 45 * cos 54.7 = 21.3 MPa

τCR cos Ø cos λ 30 σy = = cos 45 cos 54.7 σy = 73.4 MPa

Slip Motion in Polycrystals 300 mm • Stronger - grain boundaries pin deformations • Slip planes & directions (l, f) change from one crystal to another. • tR will vary from one • The crystal with the largest tR yields first. • Other (less favorably oriented) crystals yield later. Adapted from Fig. 7.10, Callister 7e. (Fig. 7.10 is courtesy of C. Brady, National Bureau of Standards [now the National Institute of Standards and Technology, Gaithersburg, MD].)

Critical Resolved Shear Stress • Condition for dislocation motion: 10-4 GPa to 10-2 GPa typically • Crystal orientation can make it easy or hard to move dislocation tR = 0 l =90° s tR = s /2 l =45° f tR = 0 f =90° s  maximum at  =  = 45º

DISLOCATIONS & σ-ε CURVES

III II I

Elastic Region: σy is the stress required to start plastic deformation Elastic Region: σy is the stress required to start plastic deformation. It is called the yield stress. From σy & on → plastic deformation starts. Stage I: dσ/dε ≈ 0 (work hardening rate) is low because only the primary slip systems are active. The slip planes are parallel to each other and only these parallel planes will slip and they do not intersect themselves, i.e. dislocations are moving along parallel planes.

Stage II: Other dislocations will start to move along intersecting planes (more than one slip system becomes active). Therefore they form barriers to one another’s motion. It becomes harder to further deform the material. This stage is known as Work Hardening Stage. Stage III: The geometry of the planes have so changed that the planes will accelerately slip and failure will occur.