CONTINUUM MECHANICS for ENGINEERS

3.2 Verify the result established in Problem 3.1 for the area elements
having normals
( )
1
nˆ = 2eˆ + 3eˆ + 6eˆ
7
1 1 2 3
( )
1
nˆ = 3eˆ − 6eˆ + 2eˆ
7
2 1 2 3
if the stress matrix at P is given with respect to axes Px 1x 2x 3 by
[ σ ij]=
⎡35
0 21⎤
⎢
⎥
⎢
0 49 0
⎥
⎣
⎢21
0 14⎦
⎥
3.3 The stress tensor at P relative to axes Px 1x 2x 3 has components in MPa
given by the matrix representation
⎡σ
11 2 1⎤
⎢
⎥
[ σ ij]=
⎢
2 0 2
⎥
⎣
⎢ 1 2 0⎦
⎥
where σ is unspecified. Determine a direction ˆn at P for which the
11
plane perpendicular to ˆn will be stress-free, that is, for which t
on that plane. What is the required value of for this condition?
Answer: , = 2 MPa
3.4 The stress tensor has components at point P in ksi units as specified
by the matrix
nˆ ( )
= 0
σ 11
1
n= ˆ ( 2eˆ eˆ 2eˆ
1− 2 − 3)
3 σ 11
[ σ ij]=
⎡−9
3 −6⎤
⎢
⎥
⎢
3 6 9
⎥
⎣
⎢−6
9 −6⎦
⎥
Determine:
(a) the stress vector on the plane at P whose normal vector is
( 1 2 3)
1
nˆ = eˆ + 4eˆ + 8eˆ
9
(b) the magnitude of this stress vector