Given the velocity field in spherical coordinates:v= (Cr +) sin de,(a) Determine the acceleration field.(b) Find the rate of deformation tensor.Using the result of Problem 1, show that for sufficiently small strainsdetF =1+ trEwhere E is the infinitesimal strain tensor. Hint: apply a Taylor series expansions to theleft-hand side. The series expansion of any scalar function f of a second order tensor T isgiven byTef(T + dT) = f(T) + IT: dT + dT :IT: dT + ...%3DOTƏTwhere dT represents a small increment relative to T. Apply this formula for f(T) = detT.%3D

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Answered: Given the velocity field in spherical…

Given the velocity field in spherical coordinates: B v = (Cr+) sin de, (a) Determine the acceleration field. (b) Find the rate of deformation tensor.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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Question

Given the velocity field in spherical coordinates:
v= (Cr +) sin de,
(a) Determine the acceleration field.
(b) Find the rate of deformation tensor.
Using the result of Problem 1, show that for sufficiently small strains
detF =1+ trE
where E is the infinitesimal strain tensor. Hint: apply a Taylor series expansions to the
left-hand side. The series expansion of any scalar function f of a second order tensor T is
given by
Te
f(T + dT) = f(T) + IT: dT + dT :
IT: dT + ...
%3D
OTƏT
where dT represents a small increment relative to T. Apply this formula for f(T) = detT.
%3D

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