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Problems
for Chapter 10
Approximate
Theories for Solids with Special Shapes:
Rods,
Beams, Membranes, Plates and Shells
10.1. Dyadic Notation
10.1.1. Let   be a Cartesian basis. Express the identity tensor as a dyadic
product of the basis vectors
10.1.2.  and  be two Cartesian bases. Show that the tensor  can be visualized as a rigid rotation (you
can show that R is an orthogonal
tensor, for example, or calculate the change in length of a vector that is
multiplied by R).
10.1.3.
Let a and b be two distinct vectors (satisfying  ).Â
Let .  Find an expression for all the vectors u that satisfy
10.1.4. Find the eigenvalues and eigenvectors of the tensor  in terms of a, b, and their
magnitudes (Don’t forget to find three
independent eigenvectors).
10.1.5.
Let . Find the condition on a and b necessary to
ensure that S is orthogonal.
10.1.6.
Let  be three linearly independent vectors. Define  to be three vectors that satisfy
and let ,
 and  denote the 27 possible dot products of these
vectors.
10.1.6.1.
Find
expressions for  in terms of vector and scalar products of
10.1.6.2.
Let  be a general second order tensor. Find expressions for ,
,
 satisfying
in
terms of  and ,
 and
10.1.6.3.
Calculate .  What does the tensor  represent?
10.1.6.4.
Express  in terms of  and appropriate combinations of ,
 and Â
10.1.6.5.
Express  in terms of  and appropriate combinations of ,
 and
10.1.6.6.
Let F denote a homogeneous deformation
gradient, satisfying .
Express F in terms of dyadic
products of  and .
10.1.6.7.
Find an
expression for  in terms of scalar products of  and dyadic products of ,
i.e. find components  satisfyingÂ
10.1.6.8.
Find an
expression for the Lagrange strain tensor  in terms of dyadic products of ,
i.e. find  satisfying ,
in terms of scalar products of  and appropriate combinations of ,
 and
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