Problem solving session for physical metallurgy
WS 2016 / 201712.12.2016
Section A:Mostly theoretical so try to find them in the book and do it yourself.
Section B:To be solved during the tutorials. (Complete solutions)
Section C:Similar to Section B that you need to practice at home
A1a)Sketch and label schematically the components of a stress tensor 2nd rank.
b)What is a stress deviator?
c)Divide a general stress tensor 2nd rank into its hydrostatic and deviatoric components (formulas).
d)Sketch and explain Mohr’s circle.
A2In a 2-dimensional referencesystem the following stress tensor is given:
a)Rotate this stress tensor by 35°.
b)Determine the principal stress tensor. How many degrees do you have to rotate ?(Answer:Θ = 26.56°)
c)Calculate the maximum shear stress. (Answer: τmax = 250 MPa)
A3a) Which vectorsclearly determine a dislocation?
b)How is the slip plane determined, by the use of these two vectors?
c)Explain the Burgers-circuit.First determinea co-ordinate system.
d)How many different types of dislocations do exist? What is the difference between edge and screw dislocations with regard to their freedom of movement?
e)Which dislocation types does a square dislocation ring consist of, if the Burgers vector lies within the plane (figure 3.12.)? In which directions do the individual dislocations move under external shear stress? (Sketch a cubic body. Draw the Burgers vector parallel to an axis. Determine the line element of the dislocation.)
What does the crystal look like when the dislocation ring has left the crystal?
f)Is it possible to producedislocation rings that consist exclusively of either edge or screw dislocations? Give an explanation.
g)What does the crystal look like when an edge dislocation ring has left the crystal?
h)How do dislocation rings develop (mixed rings,edge rings)?
i)How many atoms must condense to let a prismatic dislocation ring with a radius of r=0.6µm develop in a cubic primitive crystal (a=0.35nm)?(Answer: 9.23·105 atoms)
A4What is conservative and non-conservative dislocation motion?
A5What is the maximum possible dislocation density in a crystaland why? What is the dislocation length in 1m3 of very high deformed copper (in light years)?
(1 light year = 9.45·1015 m) (Answer: ρ = 1016 m-2, 1.058 light years)
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B1What form does Hooke’s law take in the 1-dimensional and in tensorial notation? How many components does the tensor of the elastic constants comprise? Why is the number of components actually reduced for the most materials?
B2Discuss the tensile test on a slender rod.
a)Draw a qualitative diagram of the nominal stress over the strain.
Which material constants can be extracted from this diagram?
b)Draw a qualitative diagram of the true stress over truestrain.
c)How can you convert the diagrams a) and b) into each other?
d)Explain the terms “physical hardening” and “geometrical softening”.
e)Why does the sample finally break?
B3Calculate,by use of the strain tensor, for a screw dislocation:
a)the shear stress fieldat the distance r from the dislocation core.
b)the density of energy and the total energy per length unit. Assume theor=G/(2·) for the energy of the dislocation core. The inner cut-off radius shall be b.
c)How large is the energy fraction in the elastic stress field in proportion to the total energy?
d) On the basis of these results, discuss the freedom of movement of a dislocation again.
(sample solution)
B4a)Derive a formula for calculating the Schmid-factor.
b) What is the Schmid’s shear stress law.
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C1In a tensile test a technical stress-strain diagram with a weak maximum has been plotted. Determine theuniform strain by the use of the Considère-Construction.
C2Name some typical slip systems of pure metals.
C3What is the theoretical shear strength? Derive a formula to calculate the theoretical shear strength. Calculate a rough value for the theoretical shear strength of Au (EAu=78Gpa, Au=0.44). (Answer:)
Mehran AfsharE-mail:
Institut für Metallkunde und Metallphysik
Kopernikusstr. 14Weekly doubt session: By appointment
Room E01, Tel.: 0241-80 – 26892Venue: Room E01
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