Problems for Appendix C

Index Notation for Vectors and Tensors

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

(a) Â Â Â Â (b) Â Â Â (c) Â Â Â (d)

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

(a) Â (b) Â Â Â (c) Â (d) Â Â (e) Â 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    Â Write out in full the three equations expressed by

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

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

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

C.7    Let

be two tensors.Â Â  Calculate

C.8    Let .Â  Calculate Â 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 Â Â Let Â 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 Â Â Let Â denote the components of a second order tensor and let Â denote the determinant of F.Â  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 Â Â Â Â Â Â Â

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

where Â is the secant modulus, and

(c) A.F. Bower, 2008
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