TMHL41 2014-06-11: Solution
TMHL41 2014-06-11: Solution
TMHL41 2014-06-11: Solution
1. Evaluate the following expressions involving the Kronecker delta and the permutation symbol
:
(a)
(b)
(c)
SOLUTION ----------------------------------------
(a)
(b)
(c)
2 Fill out the following table, i.e., mark which of the tensors are (by definition) symmetric and
nonsymmetric, respectively. (Note that even those which are not symmetric by definition may very
well be symmetric in particular cases!)
Symmetric Nonsymmetric
(Deformation gradient X
C (Cauchy-Green right deformation tensor) X
(Lagrange strain tensor) X
(infinitesimal strain tensor) X
(Cauchy stress tensor) X
(Rotation tensor) X
SOLUTION ----------------------------------------
SOLUTION ----------------------------------------
SOLUTION ----------------------------------------
(a) An isotropic material is a material, which can be arbitrarily rotated relative to a fixed
reference system without affecting the stress strain relation expressed in that
reference system.
(b) 2 (for instance, and or the Lam constants and ) .
( ) [ ]
Compute the normal stress and the shear stress on the plane defined by its normal vector
TMHL41 2014-06-11
SOLUTION ----------------------------------------
a he nor al stress [ ]
( ) ( ) ( )
he shear stress
[( ) ( ) ]
and so [( ) ( ) ] ( )
A line element has in the reference configuration been marked in the plane.
Compute so that the stretch will be as large as possible and compute this ax .
SOLUTION ----------------------------------------
By Spencer p. 68:
We have
( ) [ ]
Extremum search
d
sin cos cos sin sin cos
d
d sin
sin cos
d cos
[arctan ]
We must also check whether the extremes are min. or max. extremes:
d
cos sin
d
d
local in
d
d
local ax
{ d
ax sin
and can be computed from Eq. (3): sin cos
Assume that deformations are small and that the material is linearly elastic ( .
(a) Compute the Cauchy stress tensor and find any necessary conditions on the constants
and for static equilibrium to be fulfilled.
(b) Compute the principal stresses (provided that the above conditions are fulfilled).
SOLUTION ----------------------------------------
(a)
( ) [ ]
( ) ( ) [ ]
TMHL41 2014-06-11
( ) [ ]
{ {
(b)
according to Eq. (3) gives the determinant equation to solve for principal stresses:
| |
[ ]{[ ] }
[ ]
Perform the polar decomposition , i.e.,
(a) determine the components of the right stretch tensor in this coordinate system, and
(b) determine the corresponding rotation tensor .
SOLUTION ----------------------------------------
(a)
[ ][ ] [ ]
TMHL41 2014-06-11
det | |
| |
{
[ ]
eigen ector [ ]
[ ]
{[ ] [ ] [ ]
[ ]
eigen ector [ ]
[ ]
{[ ] [ ] [ ]
[ ]
eigen ector [ ]
[ ]
{[ ] [ ] [ ]
Since we now have the principal values , and and the corresponding principal directions
and of we can easily compute in the original representation:
( )
[ ][ ] [ ] [ ]
[ ]
TMHL41 2014-06-11
( ) [ ]
[ ]
[ ]