Question

The state of strain at a point is plane strain with εx = ε0, εy = –2ε0, γxy = 0, where ε0 is a positive constant. What is the normal strain along an axis oriented 45° from the positive x axis in the clockwise direction?

Answer 1

The normal strain along an axis oriented 45° from the positive x axis in the clockwise direction is -ε₀/2

Step-by-step explanation:

Given that

`\epsilon_(x)=\epsilon_(o)\\\\\epsilon_(y)=-2\epsilon_(o)\\\\\gamma_(xy)=0\\\\\theta=-45^(o)\\\\\epsilon_{x_(1)}=?`

From equation of normal strain in x direction:

`\epsilon_{x_(1)}=\epsilon_(x)cos^(2)\theta+\epsilon_(y)sin^(2)\theta+\gamma_{xy{ sin\theta cos\theta`

Substituting the values:

`\epsilon_{x_(1)}=\epsilon_(o)cos^(2)(-45)-2\epsilon_(o)sin^(2)(-45)+0\\\\\epsilon_{x_(1)}=(\epsilon_(o))/(2)-2(\epsilon_(o))/(2)\\\\\epsilon_{x_(1)}=-(\epsilon_(o))/(2)`