Chapter 16: Problem 25
At a certain point \(P\) (which you can choose to be your origin) in a continuous solid, the strain tensor is \(\mathbf{E}\). 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 \(\left(1+\epsilon_{11}\right)\) (b) Hence show that any small volume around \(P\) has changed by \(d V / V=\operatorname{tr}\) E. This shows that any two strains that have the same trace dilate volumes by the same amount. In the decomposition \(\mathbf{E}=e \mathbf{1}+\mathbf{E}^{\prime}\) \((16.88),\) the spherical part \(e 1\) changes volumes by the same amount as \(\mathbf{E}\) itself, while the deviatoric part \(\mathbf{E}^{\prime}\) doesn't change volumes at all.
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.