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At a certain point P (which you can choose to be your
Chapter 16, Problem 16.25(choose chapter or problem)
At a certain point P (which you can choose to be your origin) in a continuous solid, the strain tensor is E. Assume for simplicity that whatever displacements have occurred left P fixed and the neighborhood of P unrotated. (a) Show that the x axis near P is stretched by a factor of (1 + E11). (b) Hence show that any small volume around P has changed by dV I V = tr E. This shows that any two strains that have the same trace dilate volumes by the same amount. In the decomposition E = el + (16.88), the spherical part el changes volumes by the same amount as E itself, while the deviatoric part E' doesn't change volumes at all.
Questions & Answers
QUESTION:
At a certain point P (which you can choose to be your origin) in a continuous solid, the strain tensor is E. Assume for simplicity that whatever displacements have occurred left P fixed and the neighborhood of P unrotated. (a) Show that the x axis near P is stretched by a factor of (1 + E11). (b) Hence show that any small volume around P has changed by dV I V = tr E. This shows that any two strains that have the same trace dilate volumes by the same amount. In the decomposition E = el + (16.88), the spherical part el changes volumes by the same amount as E itself, while the deviatoric part E' doesn't change volumes at all.
ANSWER:Step 1 of 5
Part a:
The strain tensor at a point P is given by,
The derivative matrix is given by,
Where 
The separation 
Where 