MAE M256B, CEE M230B
HW #1 - Solution
Problem 1
A is a second nonsingular order tensors in a 3-dimensional space. Let
B = AA T and C = A T A.
(a) Show that B2 = ACA T and C2 = A T BA .
B2 = BB = AA T AA T = ACA T , C 2 = A T AA T A = A T BA ,
(b) Write al
More Hints on HW #5
Problem 8.1
Assume that A < X1 < 4A (instead of 5A).
The material is Neo-Hookean and the outer surface is traction free. Calculate the traction
components on the flat end sections and on the inner curved surface in the spatial
confifur
MAE 256B, CEE 230B, HW #5 with hints
Reader Problem 8.1 (modified)
Assume that A < X1 < 4A (instead of 5A).
In (c) assume that the material is of the Mooney-Rivlin type and the outer surface is
traction free. Calculate the traction components on the end s
MAE M256B, CEE M230B, HW #4
Problems 7.1, 3, 4, 5
Problem 7.5 (Hints)
W I W II W III
ij =
+
+
I ij II ij III ij
I
II
III
= ij , = I ij ij ,
= II ij I ij + ik kj
ij
ij
ij
ij
approx
= (2c1I + 3c3I 2 + c 4 II + c 4 I 2 + c5II )ij (c 2 + c 4 I + c5I )ij +
MAE M256A, CEE M230A, HW #4 Solution
Problem 7.1
B = FFT = (I + H)(I + H T ) = I + H + H T + HH T = I + 2 + s
B I = 2 + s
1
I B = trB = 3 + 2I H + Is , where = (H + H T ), s = HH T
2
a) In the linear theory
B = I + 2 + O( 2 ); I B = tr (B) = 3 + 2I + O( 2
MAE M256B, CEE M230B
HW #3-Solution
6.3
PF + H = K + UdVt
Vt
PF = K + ijv i, jdVt (Eq. 6.51), H = hdVt q i n ids
Vt
Vt
St
hdVt q i n ids + ijv i, jdVt UdVt = 0
Vt
St
Vt
Vt
1
1
1 dV
dVt
= F ij = FiI Ij , = 0 (Eq.5.19),
= dV0
J
J
J dVt
J
ijv i, jdVt = Fi
MAE M256B, CEE M230B
HW #3
Course Reader Problems 6. 3, 4, 5.
Hints:
6.3
Global Equation: PF H K UdVt
Vt
where PF K ij vi, jdVt and H hdVt q i n i ds
Vt
Vt
St
Local form:
FiI Ij vi,J G Jj 0 h (Jq i G Ii ),I 0 U 0, X R 0
6.4
Recall formulas:
[] J 1[F], cfw
HW #2 Solution
MAL
Problem 1 (5.3)
x1 = X1e t + X3 (e t 1), x 2 = X 2 + X3 (e t e t ), x 3 = X3
t
t
t
t
X1 = x1e x 3 (1 e ), X 2 = x 2 x 3 (e e ), X3 = x 3
et
J= 0
0
0
(1a)
(1b)
et 1
1 e t e t = e t > 0
0
1
Velocity field.
Material description:
V1 =
Dx1
D
MAE M256B, CEE M230B
HW #2
Problem 1. Modified 5.3 in Reader
For the given motion show that J 0 and obtain the material and spatial descriptions of the
velocity and acceleration fields.
Problem 2. Problem 5.5 in Reader
Problem 3. Modified Problem 5.6 in R
MAE M256B, CEE M230B
HW #1
Problem 1
A is a second nonsingular order tensors in a 3-dimensional space. Let
B = AA T and C = A T A.
(a) Show that B2 = ACA T and C2 = A T BA .
(b) Write all four equations above in index notation using indices i, j, k, l.
Pr