In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, hydrostatic stress, also known as volumetric stress,
is a component of stress which contains uniaxial stresses, but not shear stresses.
A specialized case of hydrostatic stress, contains
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
compressive stress
In long, slender structural elements — such as columns or truss bars — an increase of compressive force ''F'' leads to structural failure due to buckling at lower stress than the compressive strength.
Compressive stress has stress units (fo ...
, which changes only in volume, but not in shape.
Pure hydrostatic stress can be experienced by a point in a fluid such as water. It is often used interchangeably with "
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
" and is also known as confining stress, particularly in the field of geomechanics.
Hydrostatic stress is equivalent to the average of the uniaxial stresses along three orthogonal axes and can be calculated from the first invariant of the stress tensor:
Its magnitude in a fluid,
, can be given by:
:
where
is an index denoting each distinct layer of material above the point of interest,
is the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of each layer,
is the
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
(assumed constant here; this can be substituted with any
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
that is important in defining
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
), and
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
:
where the index
indicates "water".
Because the hydrostatic stress is isotropic, it acts equally in all directions. In
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
form, the hydrostatic stress is equal to
:
where
is the 3-by-3
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
.
Hydrostatic compressive stress is used for the determination of the
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describe ...
for materials.
References
{{Reflist
Continuum mechanics
Orientation (geometry)