Hydrostatic stress

Hydrostatic stress and thermodynamic pressure

In the particular case of an incompressible fluid, the thermodynamic pressure coincides with the mechanical pressure (i.e. the opposite of the hydrostatic stress): p = - \sigma_h = - \frac 1 3 \operatorname{tr} (\boldsymbol \sigma)

In the general case of a compressible fluid, the thermodynamic pressure p is no more proportional to the isotropic stress term (the mechanical pressure), since there is an additional term dependent on the trace of the strain rate tensor:

p = - \frac 1 3 \operatorname{tr} (\boldsymbol \sigma) + \zeta \operatorname{tr} (\boldsymbol \epsilon)

where the coefficient \zeta is the bulk viscosity. The trace of the strain rate tensor corresponds to the flow compression (the divergence of the flow velocity):

\operatorname{tr} (\boldsymbol \epsilon) = \operatorname{tr} \left(\frac 1 2 (\nabla \mathbf u + (\nabla \mathbf u)^T) \right) = \nabla\cdot\mathbf{u}

So the expression for the thermodynamic pressure is usually expressed as:

p = - \sigma_h + \zeta \nabla\cdot\mathbf{u} = \bar p + \zeta \nabla\cdot\mathbf{u}

where the mechanical pressure has been denoted with \bar p. In some cases, the second viscosity \zeta can be assumed to be constant in which case, the effect of the volume viscosity \zeta is that the mechanical pressure is not equivalent to the thermodynamic pressure as stated above. \bar{p} \equiv p - \zeta , \nabla \cdot \mathbf{u} , However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming \zeta = 0. The assumption of setting \zeta = 0 is called as the Stokes hypothesis. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect.

Potential external field in a fluid

Its magnitude in a fluid, \sigma_h, can be given by Stevin's Law:

:\sigma_h = \displaystyle\sum_{i=1}^n \rho_i g h_i

where

• i is an index denoting each distinct layer of material above the point of interest;
• \rho_i is the density of each layer;
• g is the gravitational acceleration (assumed constant here; this can be substituted with any acceleration that is important in defining weight);
• h_i is the height (or thickness) of each given layer of material.

For example, the magnitude of the hydrostatic stress felt at a point under ten meters of fresh water would be

:\sigma_h = \rho_w g h_w =1000 ,\text{kg m}^{-3} \cdot 9.8 ,\text{m s}^{-2} \cdot 10 ,\text{m} =9.8 \cdot {10^4} \text{ kg m}^{-1} \text{s}^{-2} =9.8 \cdot 10^4 \text{ N m}^{-2}

where the index w indicates "water".

Because the hydrostatic stress is isotropic, it acts equally in all directions. In tensor form, the hydrostatic stress is equal to

:\sigma_h \cdot I_3 = \sigma_h \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right] = \left[ \begin{array}{ccc} \sigma_h & 0 & 0 \ 0 & \sigma_h & 0 \ 0 & 0 & \sigma_h \end{array} \right]

where I_3 is the 3-by-3 identity matrix.

Hydrostatic compressive stress is used for the determination of the bulk modulus for materials.

References

• Stress tensor
• Volumetric strain
• Deviatoric stress tensor
• Flow velocity
• Pressure
• Bulk viscosity
• Isotropy

Notes

References

1. Megson, T. H. G. (Thomas Henry Gordon). (2005). "Structural and stress analysis". Elsevier Butterworth-Heineman.
2. Soboyejo, Winston. (2003). "Mechanical properties of engineered materials". Marcel Dekker.
3. Landau & Lifshitz (1987) pp. 44–45, 196
4. White (2006) p. 67.
5. Stokes, G. G. (2007). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.
6. Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington.