Stress problem, units? (some calculation inside)

[n13]

Homework Statement

The stress of a particular material, σ, is a function of x and y, where x is stress in the x-direction and y is stress in the y direction. the stress function is given by:

σ(x,y)= $e^{y}$ ln(2x+3y)

when x=2cm @ 3cm/s​

y=3cm @ 1cm/s​

Homework Equations

$\frac{dσ}{dt}$=$\frac{∂σ}{∂u}\frac{dx}{dt}+\frac{∂σ}{∂y}\frac{dy}{dt}$

∴$\frac{dx}{dt}$ = 3cm/s
and $\frac{dy}{dt}$ = 1cm/s

The Attempt at a Solution

Homework Statement

I have:

$\frac{∂σ}{∂y}$ = $\frac{3}{2x+3y}$

and

$\frac{∂σ}{∂x}$ = $\frac{2}{2x+3y}$

Homework Equations

Im not sure if these partial differentiations are correct.

The Attempt at a Solution

Homework Statement

carrying on asif they were:

$\frac{dσ}{dt}$= $\frac{3}{2x+3y}$ (3) + $\frac{2}{2x+3y}$ (2)

sub rates in for x and y:

$\frac{dσ}{dt}$= $\frac{9}{2(3)+3(1)}$ + $\frac{4}{2(3)+3(1)}$ (2)

Homework Equations

so summing up i end up with:

$\frac{dσ}{dt}$= 1.44

I was wondering what the units would be? considering that the original problem states stress, I can't figure out what the units are?

 

[mighty2000]

The Attempt at a Solution

The units for stress are typically in pascals (Pa) or newtons per square meter (N/m^2). In this case, since the stress function is given in terms of x and y, the units for x and y would also need to be included in the final answer. So the units for the derivative of stress with respect to time would be Pa/(cm/s).