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.           Find expressions for  in terms of vector and scalar products of           Let  be a general second order tensor.  Find expressions for , ,  satisfying


in terms of  and ,  and           Calculate .   What does the tensor  represent?           Express  in terms of  and appropriate combinations of ,  and           Express  in terms of  and appropriate combinations of ,  and           Let F denote a homogeneous deformation gradient, satisfying . Express F in terms of dyadic products of  and .           Find an expression for  in terms of scalar products of  and dyadic products of , i.e. find components  satisfying           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
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