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Problems
for Appendix C
Index
Notation for Vectors and Tensors
A.
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)  Â Â Â Â (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)  Â
(f) Â Â Â Â Â Â Â Â Â Â Â (g) Â Â (h) Â Â Â Â Â (i) Â Â Â Â (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 Â Â The strain
energy density for a hypoelastic material is given by
where
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 Â Â The strain
energy density of a hyperelastic material with a Neo-Hookean constitutive
relation is given by
where
Show that
You may use the solution to
problem C.9
C.14 Â Â A
hypoelastic material has a stress-strain relation given by
where
       Â
and  is the slope of the uniaxial stress-strain
curve at . Show that
where  is the secant modulus, and
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