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 ). 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
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 satisfying
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
(c) A.F. Bower, 2008