strain fields

[logistic_guy]

Determine whether the following strain fields are possible in a continuous material:

\(\displaystyle \bold{a.} \ \begin{bmatrix}c(x^2 + y^2) & cxy \\cxy & y^2 \end{bmatrix} \)

\(\displaystyle \bold{b.} \ \begin{bmatrix}cz(x^2 + y^2) & cxyz \\cxyz & y^2z \end{bmatrix} \)

Here \(\displaystyle c\) is a small constant, and it is assumed that \(\displaystyle \varepsilon_{z} = \gamma_{xz} = \gamma_{yz} = 0\).

 

[logistic_guy]

To solve this problem, we start by introducing the two-dimensional strain matrix.

[imath]\begin{bmatrix}\varepsilon_x & \frac{1}{2}\gamma_{xy} \\[6pt] \frac{1}{2}\gamma_{xy} & \varepsilon_y \end{bmatrix} [/imath]

This matrix along with the two-dimensional strain partial differential equation will be handy to solve the OP problem.