API ABDUL KALAM TECHNOLOGICAL UNIVERSITY
FIRST SEMESTER M.TECH DEGREE EXAMINATION, DECEMBER 2015
Mechanical Engineering
(Machine Design)
01ME6105 Continuum Mechanics

Max. Marks : 60 Duration: 3 Hours
Answer any two full questions from each module.

Part A (Modules 1 & 11) - Max marks:18

1.(2) Given a continuum, where the stress state is known at one point and is represented

1 10 by the Cauchy stress tensor
components % =t 1 0 Pa. find the principal stresses

002

and principal directions (6 marks)
(0) Show that (1) 539५9 = v3 (1) 43 (1) 6126134൮
(3 marks)
1423
2. (a) Prove the identity 6jmékn _ ௭9௭ (5 marks)
5 6 7
(b) The Cauchy stress tensor at point P is given by Gij = 6 8 9(iPa.O brat
n
7 9 2
the deviatoric and volumetric parts of thc tensor. (4 marks)
3. (a) Prove the vector identity X (v X w) ಎ (ப. ~ (५. v)w (5 marks)

(b) The stress Sla:e al one point is represented by the Cauchy stress components

a;j — aaaco. where 0. 0, c are constants and ௦ is the value of the stress.
|)" 099
Determine [he constants such that the traction vector on the octahedral plane is

zero. (4 marks)

Part B (Modules 111 & IV) - Max marks:18
4. (a) Given the motion of a body to be = Xi +0.2tX251i, for a temperature field given by
0 = 2x, + (02, find the material description of temperature and rate of change of
temperature ofa particle, which at time t = O was at the place (0, 1,0).
(5 marks)
(b) Deduce the equilibrium equations from linear momentum principle. (4 marks)
5. (a) Obtain the infinitesimal strain tensor and the infinitesimal spin tensor for the
following displacement field ; 72," , = x,x2, O (5 marks) (0) Prove the symmetry of
stresses (Jij = Oji using principle of conservation of angular momentum (4 marks)