Description
1. [20 points] (Literature survey) Select one of the following papers and summarize (i) the problem;
(ii) objective; (iii) approach; (iv) findings; and (v) conclusions in 200 words.
– R.W. Ogden, Large Deformation Isotropic Elasticity – On the Correlation of Theory and Experiment for
Incompressible Rubberlike Solids, Proc. R. Soc. Lond. A, vol. 326 no. 1567 565-584, 1972.
– Zhengyou Liu, Xixiang Zhang, Yiwei Mao, Y. Y. Zhu, Zhiyu Yang, C. T. Chan, Ping Sheng, Locally Resonant Sonic
Materials, Science, Vol. 289 no. 5485 pp. 1734-1736, 2000.
– S. Cai, D. Breid, A.J. Crosby, Z. Suo, J.W. Hutchinson, Periodic patterns and energy states of buckled films on
compliant substrates, Journal of the Mechanics and Physics of Solids, Volume 59, Issue 5, Pages 1094–1114, 2011.
– James C. Weaver et al., The Stomatopod Dactyl Club: A Formidable Damage-Tolerant Biological Hammer, Science,
336 (no. 6086), pp. 1275-1280, 2012.
2. [60 points] The displacement field in a
homogeneous, isotropic circular shaft (Radius R)
twisted through angle
at one end is given by
3 3
1 1 2
3 3
2 1 2
3
[cos 1] sin
sin [cos 1]
0
x x
u x x
L L
x x
u x x
L L
u
= − −
= + −
=
2.1. Calculate the matrix of components of the deformation gradient tensor.
2.2. Calculate the matrix of components of the Lagrange strain tensor. Is the strain tensor a
function of
x3
? Why?
2.3. Find an expression for the increase in length of a material fiber of initial length dl, which is on
the outer surface of the cylinder and initially oriented in the
e3
direction.
2.4. Show that material fibers initially oriented in the
e1
and
e2
directions do not change their
length.
2.5. Calculate the principal values and directions of the Lagrange strain tensor at the point
x1
= a = L /10, x2
= 0, x3
= 0
under
a = 5
degrees. Hence, deduce the orientations of the
material fibers that have the largest extension and contraction in length (hint: Use ‘eig’
function in Matlab).
e1
e2
e3
L
AA 530: SOLID MECHANICS Out: Oct. 5, 2021
HW #1 Due: Oct. 14, 2021
2
2.6. Calculate the components of the infinitesimal strain tensor. Show that, for small values of
, the infinitesimal strain tensor is identical to the Lagrange strain tensor, but for finite
rotations the two measures of deformation differ.
2.7. Use the infinitesimal strain tensor to obtain estimates for the lengths of material fibers initially
oriented with the three basis vectors. Where is the absolute error in this estimate greatest?
When
R = L /10
, how large can
be before the absolute error in this estimate reaches a
10% strain value?
2.8. [Bonus problem: 20 points] Visualize displacement fields of the given problem using
Matlab (L = 1 m, R = 0.1 m,
a = 5
degrees).
3. [20 points] Consider a strain state described in rectangular Cartesian coordinates:
e xx
= k x
2
+ y
2
( ), e yy
= k y
2
+ z
2
( ), e xy
= k ‘ xyz
e xz
=e yz
=e zz
= 0
where k and k’ are small constants. Is this a possible state of strain for a continuum?
