Linear Elasticity: Tutorial Sheet 1

Linear Elasticity: Tutorial Sheet 1
Question 1
A motor car tyre is constructed with inextensible cords such that no deformationperpendicular to the length of the tyre is possible, i.e.
The tyre is under uni-axial tension in theOx1direction. The effect of the inexten-sible cords is to introduce an extra force acting to ensure no deformation in thex2direction.(a) Suggest a suitable displacement field and components of the stress tensorwhich will enable you to model this situation. Hence, obtain the componentsof the stain tensor.(b) Use the stress-strain relationship to write the unknowns in your displacementfield in terms of the Lammé constants and the uni-axial tension T.

Question 2
A cantilever beam with rectangular cross-section occupies the region a ≤ x1 ≤ a, h ≤ x2 ≤ h, 0≤ x3 ≤ l. The endx3=lis built in and the beam is bent by aforcePapplied at the free endx3= 0 and acting in thex2direction. The stresstensor has components
(a) Given that the body is in equilibrium (i.e.u=), show that the equation ofmotion with no body forces is satisfied provided 2B+C=.(b) determine the relationship betweenAandBif no traction acts on the sidesx2=±h.(c) express the resultant force on the free endx3= 0 in terms ofA,BandCand hence, withaandbshow thatC= 3P4ah3

Question 3
The components of the stress tensorTijat a certain point in an elastic body aregiven by Given that the stress vectortin plane through this point ist= (0,0, P), whereTis a constant, determine a unit vectornon this plane, and the value ofT33.

Question 4
A three-dimensional elastic body is in a state of plain strain such that all deforma-tions are independent ofx3and lie in planes parallel tox3= 0. The body is linearlyelastic with Lame’ http://www.ukassignment.org/essaygs/ moduliλandμ.(a) Write down the components of the strain tensoreijin the body. In particular,indicate any components that are zero.(b) Use the constitutive lawto deduce the corresponding form of the stress tensor in terms of the non-zerocomponents of the strain tensor.(c) Can the stresses act in thex3direction?

Question 5
Show that in a homogeneous, isotropic, linearly elastic body the principal axes ofthe stress and strain tensors coincide.