HOMEWORK 9 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I , FALL 2011
1. P ROBLEM 1
(a) Given T = (tr E) I + 2 E, we nd that tr T = (3 + 2 ) tr E, from which it follows that
tr T
tr E =
3 + 2
Substituting this relation into T produces
T=
Solving for E y
HOMEWORK 7
MECE E6422 INTRODUCTION TO ELASTICITY I , FALL 2011
1. P ROBLEM 1
(a) Solve for E from the expression T = (tr E) I + 2 E and make use of the relations between
, and EY , to show that
E=
1
[(1 + ) T (tr T) I]
EY
(1.1)
(b) Evaluate the complianc
HOMEWORK 8 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I , FALL 2011
1. P ROBLEM 1
(a) Using indicial notation,
(tr A)
Akk
=
= ki k j = i j = (I)i j
A
Ai j
ij
tr A2
(Akl Alk )
=
A
Ai j
ij
Akl
Alk
Alk + Akl
Ai j
Ai j
= ki l j Alk + Akl li k j
HOMEWORK 7
MECE E6422 INTRODUCTION TO ELASTICITY I , FALL 2011
1. P ROBLEM 1
According to the Cayley-Hamilton theorem, a second-order tensor should satisfy its own characteristic equation, thus
A3 I1 A2 + I2 A I3 I = 0
(1.1)
where
1
2
2
I1 = tr A, I2 =
(t
HOMEWORK 7 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I AND FALL 2011
P ROBLEM 3.17
Consider the motion given by
x = X + X1 k e1
= X1 (1 + k) e1 + X2 e2 + X3 e3
Let d X(1) = dS1 / 2 (e1 + e2 ) and d X(2) = dS2 / 2 (e1 + e2 ).
(a) To nd the deformed el
HOMEWORK 7
MECE E6422 INTRODUCTION TO ELASTICITY I , FALL 2011
P ROBLEM 1
Solve Problem 3.17 in the textbook.
P ROBLEM 2
Solve Problem 3.18 in the textbook.
P ROBLEM 3
In linear algebra, we learn that for an arbitrary real tensor T and vector v,
f = vTv
i
HOMEWORK 6 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I AND FALL 2011
P ROBLEM 3.4
Consider
2
x1 = X2 t 2 + X1
x2 = kX2t + X2
x3 = X3
(a) At t = 0, A (0, 0, 0), B (0, 1, 0), C (1, 1, 0), D (1, 0, 0). At t = 2, A (0, 0, 0), B (4 , 2k + 1, 0),
C (4 + 1,
HOMEWORK 5 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I AND FALL 2011
1. P ROBLEM 2.65
2
Consider = x1 + 3x1 x2 + 2x3 .
(a) The unit vector normal n to the surface of constant is obtained from
grad
n=
.
|grad |
Thus, since
grad =
ei = (2x1 + 3x2 ) e1
HOMEWORK 4 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I AND FALL 2011
1. P ROBLEM 2.59
Given
(a) The characteristic equation is
540
[T] = 4 1 0
003
det (T I) = 3 + 7 2 + 9 63 = ( 7) ( 3) ( + 3) = 0
The principal values are 1 = 7, 2 = 3 and 3 = 3. The
HOMEWORK 3 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I AND FALL 2011
1. P ROBLEM 2.39
Given
1 5 5
[T] = 5 0 0
5 0 1
nd T11 , T12 and T31 with respect to cfw_e where
i
e2 + 2e3
1
= (e2 + 2e3 )
|e2 + 2e3 |
5
e2 = e1
1
e = e e = (2e2 + e3 )
3
1
2
5
e
HOMEWORK 2 SOLUTION
MECE E6422 INTRODUCTION TO ELASTICITY I AND FALL 2011
1. P ROBLEM 2.19
Given that
show that T is not a linear transformation.
Ta =
a
|a|
Proof. As a linear transformation T would need to satisfy T ( a) = T a for any vector a.
However,