EMA 6715 Mid-term Examination [Part 1]

March 5, 2003 Name __________________________________

Closed book examination.

If you think that one or more of your answers needs clarification, please write on reverse side citing the question number.

1. The Eulerian infinitesimal strain components, e_ij are related to the displacement and position vectors by the following equation: ___________________________________.

2. The components of a displacement vector are u₁ = 0.004x₁ – 0.16x₂, u₂ = 0.024x₂, u₃ = 0.025 x₃

a. What is the value for e₁₁? ________________________________________

b. What is the value for e ₁₂ ________________________________________

c. What is the value for w₂₁ ________________________________________

d. What is the value for DV/V _______________________________________

e. One of the principal strain values is _________________________________

3. Answer the following questions:

a. a_klb_klmn is a __________________________________________ order tensor.

b. Simplify d_kl d_{kl __________________________________________________________________________}

c. Give 2 properties of symmetric second order tensors ______________________________________________________________

4. For a body in tension, the deviatoric stress tensor has the following components:

5. To solve 2D elastic problems the Airy stress function is often used. This function must satisfy the boundary conditions and the _______________________ equation. This latter equation combines Hooke’s Law with the __________________ and ______________ equations.

6. For a cubic crystal, the stiffness constants are: c₁₁=c₂₂=c₃₃, c₁₂=c₁₃=c₂₃ and c₄₄=c₅₅=c₆₆, the rest = 0. The compliance constants, s_ij have the same symmetry. Write the general form of Hooke’s Law for the following components in terms of the elastic constant:

a. s₂ = _________________________________________________________________________

b. s₅ = _________________________________________________________________________

c. e₃ = _________________________________________________________________________

d. If the material is tested in tension (s₁), then s₁ = ______________________________________

7. The maximum Young’s modulus for a cubic crystal is either along _________ or _________ directions. In terms of compliance constants, Young’s modulus for a principal direction is ____________________ and Poisson’s ratio for a principal direction is ______________________________.

8. The value of the bulk modulus for a material depends on the ____________________ of the interatomic potential at the _________________________________ interatomic spacing.

9. Which of the following has the lowest theoretical fracture stress? TiC, TiN, TiO₂? _________________.

10. Which of the following has the largest bulk modulus, Na Cl, NaF, Na I, or NaBr? ________________.

Name _______________________________________________________________-

11. For alkali halides, the bulk modulus ________________________ with increasing equilibrium spacing or with _________________________________ volume per ion pair.

12. Which equations (by name) can be used to bound elastic constants of isotropic elastic constants of particulate composites without any assumption of particle shape? ______________________________

13. The Griffith criterion for failure considers the total energy, U, of a cracked body with changes in crack length, c. The failure criterion based on Griffith’s approach is usually written as: _____________________.

14. The crack extension force or strain energy release rate depends on the work done by the applied loads, W, and the change in the elastic energy of the system, U_E, such that ________________________[equation].

15. The stress intensity factor describes the _______________ of the stresses near a crack tip and is related to the applied stresses by K = ___________________________.

16. Name three (non-indentation) techniques that are used to measure fracture toughness ______________________________________________________________________________________.

17. For an elliptically shaped crack, the stress intensity factor is a minimum at the end of the ______________ axis.

18. In a beam subjected to bending, the _________________ are dependent on the material properties and the _____________________ are independent of the material properties.

19. A plate that is thick can most like be analyzed using plane _______________ [stress or strain].

20. The stress concentration factor is ______________ for a spherical pore than a circular hole.

21. If a particle in a matrix has a thermal expansion coefficient greater than that of the matrix, then cracks can form upon cooling. If these cracks form, where are the possible locations? ____________________________

22. What is the stress state of the particle in Question 21? _______________________________________

23. What is the stress state in the matrix of Question 21? ________________________________________

24. The theoretical strength of a material can be estimated from a sinusoidal stress-displacement function in which the period is l/2 where l is the wavelength of the sine function. The integration of the stress-displacement curve from 0 to l/2 determines the surface energy. Why is the integration from 0 to l/2 [ as opposed to l/4]? ________________________________________________________________________________________________________________________________________________________________________________

25. Which of the following type of defects that can lead to failure is least severe: pores, inclusions or surface cracks? _______________________________________________________________________________