# I What is physicial interpretation of deviatoric strain rate

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1. Sep 8, 2016

### Atr cheema

How these properties are related to velocity fluid. The https://postimg.org/image/674a6sw4t/ https://postimg.org/image/674a6sw4t/ figure shows an area of earth's mantle where upwelling of hot semi-liquid mantle is occurring in middle and then two downwelling currents on two sides (forming convection cells). The figure shows deviatoric strain rate (left) and its second invariant (right)

2. Sep 8, 2016

### Staff: Mentor

The deviatoric strain rate is equal to the total strain rate minus the isotopic volumetric strain rate. So it "factors out" the rate of volumetric deformation. The second invariant of the strain rate tensor is usually regarded as a scalar measure of the rate of shear.

3. Sep 8, 2016

### Atr cheema

Can you please elaborate what will remain if we subtract isotropic strain rate from total strain rate tensor? I am having difficulty in perceiving it in a physical sense.
Is there a difference between invariant of deviatoric strain rate tensor and 'strain rate tensor'? I am worrying as you used just 'invariant of strain rate tensor' while I read it like 'second invariant of deviatoric strain rate'.

4. Sep 8, 2016

### Staff: Mentor

If the strain rate were purely volumertic (isotropic) with no shear, the deviatoric part of the rate of stain tensor would be zero.

Regarding scalar shear rate, I meant to say 2nd invariant of the deviatoric strain rate tensor. The nominal scalar shear rate is often taken as being proportional to the square root of the 2nd invariant of the deviatoric strain rate tensor. If I remember correctly, the constant of proportionality is $\sqrt{2}$. You can work that out yourself by considering simple shear flow.

5. Sep 8, 2016

### Staff: Mentor

Regarding the link you provided, what are the thermal boundary conditions? What is the nature of the flow?

6. Sep 8, 2016

### Atr cheema

The x and y components of velocity field in the modeled area are as; $v_x = v_x0 sin \left (2 \Pi \frac x W \ right) cos \ left ( \Pi \frac y H \ right )$
$$v_y, = v_y0, cos \left (2 \Pi \frac x W \ right) sin \ left ( \Pi \frac y H \ right ),$$
W and H are width and height of modeled area. vxo and vyo are constants. Here is complete problem

7. Sep 8, 2016

### Staff: Mentor

Have you drawn the diagram showing the velocity vector as a function of position within the box? If so, please show us. To understand why the strain rates have the values shown, you need to understand the velocity field.

Chet

Last edited: Sep 8, 2016
8. Sep 8, 2016

### Staff: Mentor

I see that you have labelled the left figure as $\epsilon _{xy}$. Why do you think that this is the deviatoric strain rate? Also, let's see the equation you used for calculating the 2nd invariant.

9. Sep 9, 2016

### Atr cheema

Thank you very much for your interest. Here is the velocity field diagram.
It is because I used the equation for deviatoric strain rate to draw this diagram. I used following code of MATLAB to draw this.
Code (Text):
for i=1:1:31
for j=1:1:31
dvxdy(i,j)=vx0*pi/ysize*sin(pi*x(j)/xsize*2)*sin(pi*y(i)/ysize);
dvydx(i,j)=-vy0*pi/xsize*2*sin(pi*y(i)/ysize)*sin(pi*x(j)/xsize*2);
eps1xy(i,j)=1/2*(dvxdy(i,j)+dvydx(i,j));
end
end
It will be very nice of you, if you can help me interpret/justify these images.

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