Structural Mechanics CEE 471

CEE471 Fall 2015 Homework 1 Due on Wednesday September 9th at 2:00 pm. Write your name and your ID on the first page. 1) Compute the numerical values of the following expressions

where 1 0 1

))) u ( , , )

ii

rs sr

i is sj j

a

b

c

u u =

2) Making use of indicial notation, show that

1 1 2 2 3 3i iu v u v u v u v u v

3) Making use of indicial notation, show that

16

detijk pqr ip jq kr

A A A A

4) Making use of indicial notation, show that

ijk pqk ip jq iq jp

5) If ij

T are the components of an arbitrary second-order tensor, show that ijk jkT are the

components of a first-order tensor. Deduce that ijT is symmetric if and only if 0

ijk jkT .

6) If 1 trI (T) = T , 2 22

12

tr trI (T) = ( T) T , and 3 detI (T) = T are the three principal

invariants of an arbitrary second-order tensor T , show that

1 21

3

II

I (T)

(T ) =(T)

, 1 1

23

II

I (T)

(T ) =(T)

, 1

33

1I

I(T ) =

(T)

7) Make use of the Cayley-Hamilton theorem to show that

3 2 31 3 26

det tr tr tr tr T = ( T) ( T)( T ) T

where T is an arbitrary second-order tensor. 8) Let T be an arbitrary second-order tensor. Show that

1 21 2

3

1I I

I T T (T)T (T)I

(T)

Show further that nT , where n is any positive or negative integer, is expressible in terms of linear combinations I , T , and 2T multiplied by coefficients that are invariants of T .

9) Relative to certain basis, the second-order tensor T has components

1 1 01 3 4

0 4 2T

Find the eigenvectors and eigenvalues of T . Show that the three principal invariants of T are the same in the given basis as in the basis defined by its eigenvectors (i.e., the principal axes).

10) Two symmetric second-order tensors are said to be coaxial if their principal axes (i.e., their eigenvectors) coincide (in some order). Prove that S and T are coaxial if and only if

ST TS