Problems for Appendix C


Index Notation for Vectors and Tensors






C.1    Which of the following equations are valid expressions using index notation?  If you decide an expression is invalid, state which rule is violated.

(a)     (b)    (c)    (d)  




C.2    Match the meaning of each index notation expression shown below with an option from the list

(a)  (b)    (c)  (d)   (e)   
(f)            (g)   (h)      (i)     (j)  

(1) Product of two tensors

(2) Product of the transpose of a tensor with another tensor

(3) Cross product of two vectors

(4) Product of a vector and a tensor

(5) Components of the identity tensor

(6) Equation for the eigenvalues and eigenvectors of a tensor

(7) Contraction of a tensor

(8) Dot product of two vectors

(9) The definition of an orthogonal tensor

(10) Definition of a symmetric tensor




C.3     Write out in full the three equations expressed by




C.4    Let a, b, c  be three vectors.  Use index notation to show that





C.5    Let  and  be tensors with components  and .  Use index notation to show that





C.6    Let ,  and  be tensors with components ,  and .  Use index notation to show that





C.7    Let


be two tensors.   Calculate  




C.8    Let .  Calculate  and  




C.9    The stress-strain relations for an isotropic, linear elastic material are


Calculate the inverse relation giving stresses in terms of strains.





C.10   Let  denote a symmetric second order tensor, and let


Show that






C.11   The strain energy density for a hypoelastic material is given by




Show that the stresses follow as



C.12   Let  denote the components of a second order tensor and let  denote the determinant of F.  Show that





C.13   The strain energy density of a hyperelastic material with a Neo-Hookean constitutive relation is given by




Show that


You may use the solution to problem C.9




C.14   A hypoelastic material has a stress-strain relation given by




and  is the slope of the uniaxial stress-strain curve at .  Show that


where  is the secant modulus, and  





(c) A.F. Bower, 2008
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