CAM 397, EM 397 : Nonlinear Static and Dynamic Finite Element Analysis (with
Particular Emphasis on Solids and an Introduction to Isogeometric Analysis)
Unique Number: 13885 (EM 397), 63040 (CSE397)
Class Schedule: TuTh 2:00 pm -- 3:30 pm
Location: BUR136
Instructor:
Thomas J.R. Hughes
hughes@ices.utexas.edu
http://www.ices.utexas.edu/~hughes/
HW 11: Due May 5, 2016
1. Fill in the blanks
∂2 Φ
a.
= AiIjJ = ______ C____ ______+ ______ S____
∂FiI ∂FjJ
b. aijkl = cijkl + σ jlδ ik = ______ A____
d
⎛ ∂φ ⎞
2. Let J = det F(φ ) = det ⎜
. Calculate
J(φ + εΔu) .
⎟
⎝ ∂X ⎠
dε
ε =0
3. Consider an n × n matrix Tε such that there exists a unique inverse Tε−1 for
all ε ∈(-δ ,δ ) for some δ >0. Determine:
d −1
a.
Tε
dε
d
b.
det Tε
dε
d
c.
cofTε
dε
d
d.
tr ( Tε2 )
dε
d
e.
tr ( (devTε )2 )
dε
4. Let
Φ(E) :=
λ
1
⎛λ
⎞
J(E)2 − 1) − ⎜ + µ ⎟ ( ln J(E)) + µ ( trC(E) − 3)
(
⎝
⎠
4
2
2
where C(E) = 2E − I and J(E) = det F(E) = + ( det C(E))
1/2
= + ( det(2E + I)) .
1/2
Note: Φ is a polyconvex function of F.
Determine:
a. S IJ
∂S IJ
b. C IJKL =
∂E KL
c. PiI
d. σ ij
e. cijkl
f. A iIjJ
5. Derive n j da = J(F −T ) jJ N J dA
Hint: The following result is helpful: Let U and V be two vectors and M a matrix,
then
MU × MV = (det M)M −T (U × V)
where × denotes cross product.