Question

Defects and Slip

Chromium metal has a metallic atomic radius of 0.200 nm. It shows plastic deformation via slip along the <111> directions.

a) What is the planar density (atoms/m2) for its densest family of planes? What is the magnitude of the Burgers vector for the dislocations that promote slip in chromium metal?

c) A chromium ingot has tensile stress applied at a 25° angle to the slip plane normal and a 63° angle to the slip direction. It experiences a resolved shear stress of 25 MPa. What is the total tensile stress applied to the ingot? 

d) What is the maximum resolved shear stress, in any direction, for this applied tensile stress? 

e) It is observed that polycrystalline chromium has a higher tensile stren than single crystalline chromium. Propose a reason for this difference in tensile strength.

Answer 1

  a)

Chromium (Cr) has a BCC crystal structure.

Slip along <111> directions suggests a BCC system, corresponding to {110}, <111> slip.

Therefore: a?3 =4r

a= (4 x 0.200 x 10^-9) / (?3)

a= 4.61 x 10^-10 m.

Densest planes are {110}, so we find: (2 atoms) / (a² ?2)= 6.63 x 10¹⁸ atoms/m²

b)

Expressions for Burgers vectors for FCC and BCC crystal structures are of form b =(a/2) (uvw) where a is the unit cell edge length. Also, since the magnitudes of these Burgers vectors may be determined from the following equation: |b| = (a/2) {(u²+v²+ w²)}^(1/2)

Therefore, the values for u, v, and w, in are 1, 1, and 1, respectively. Hence, the magnitude of the Burgers vector for Cr is

|b|=(4.61 x 10^-10)/2 * {(1²+1²+1²)}^(1/2) = 4 x 10^-10 m

c) Critical resolved shear stress (CRSS) is the component of shear stress, resolved in the direction of slip, necessary to initiate slip in a grain. Resolved shear stress (RSS) is the shear component of an applied tensile or compressive stress resolved along a slip plane that is other than perpendicular or parallel to the stress axis. The RSS is related to the applied stress by a geometrical factor, m, typically the Schmid factor.

?_rss= ?_appm = ?_app(Cos? Cos?)

where ?_app is the magnitude of the applied tensile stress, ? is the angle between the normal of the slip plane and the direction of the applied force, and ? is the angle between the slip direction and the direction of the applied force.

Putting up given values,

25=?_app(cos 25 *Cos 63)

?_app=60.75 MPa

e)

The polycrystalline material is aggregate of several crystals or grains The boundary between the grain is the grain boundary across which the orientation of the crystal changes.The point at which three boundaries meet is called the triple junction.A general relationship between mechanical properties and grain size is given by the Hall-Petch equation ?₀=?_i+kd^(-1/2) Where ?₀ is the yield strength, d is the grain size and ?_i and k are material dependent constants.Finer grain size means more grain boundaries or higher grain boundary area per unit volume. Deformation in metals takes place by dislocation motion and grain boundaries act as obstacles to dislocation motion. Hence, the presence of more grain boundaries (finer grain size) will increase the resistance to deformation and enhance the strength. The single crystal has only one grain or crystal and hence, no grain boundaries.Hence polycrystalline chromium has a higher tensile strength than single crystalline chromium.