8 Let E denote the rate-of-strain tensor, which is the symmetric part of the velocity gradient…

8 Let E denote the rate-of-strain tensor, which is the symmetric part of the velocity gradient tensor. Is it true that ?u : ?u = E : E ? (64) Show whether this equation is true or false (i.e. provide enough details to justify your answer). Recall: A : B = AijBji.
Some tensorial operations and more on the idea of constitutive equations: (a) Consider the constitutive equation for a complex fluid in the form (using index notation) Tij = -pdij + Aijkl?luk, where ?luk = ?uk ?xl are the components of the velocity gradient tensor. Take as the definition of the “pressure” p = – 1 3 Tii as is usual in mechanics. Hence, first conclude Aiikl = 0. In the absence of body couples we expect the tensor Aijkl (or A) to be symmetrical in the indices i and j. Also, we expect the state of stress to be independent of a rigid rotation, so the stress-rate-of-strain relation involves the rate of strain tensor E and Aijkl is symmetric in k and l. How many independent components of Aijkl are there? What additional information do you need to reduce this number further?