Answer
# 23 |
Pressure (symbol: p) is the force per unit area applied on
a surface in a direction perpendicular to that surface.
Mathematically:
where:
p is the pressure
F is the normal force
A is the area.
Pressure is transmitted to solid boundaries or across
arbitrary sections of fluid normal to these boundaries or
sections at every point. It is a fundamental parameter in
thermodynamics and it is conjugate to volume.
Moments is the scientific name given to someting that turns
on a pivot (fulcrum). The pressure of the moment can also
be measured using a similar calculation to one given
earlier in this article. The thing to consider or the
relationship is that the further away the load (effort
force) is from the pivot the larger the force. For example,
a door, it has its handle (effort force) right at the end
furthest away from the hinges (pivot). This allows the door
to move with ease. The calculation used to measure force of
moments is:
Distance from Pivot (mm/cm/m/km/M) X Force (N) =
Example: A door has its handle 70cm away from its pivot and
the force exerted on the door is 8 (N)
Distance from Pivot (70) X Force (8) = 560 Ncm2
In the human body there are several types of pressure
receptor. Baroreceptors monitor blood pressure in the
carotid arteries, aortic arch and right atrium of the
heart. Mechanoreceptors are part of the somatosensory
system and are present in the dermis of the skin and in
deeper tissues. They respond to different forms of touch
and pressure; the main types of mechanoreceptor are
Pacinian corpuscles, Meissner's corpuscles, Merkel cells
and Ruffini endings.1 Pa = 1 N/m² = 10−5 bar = 10.197×10−6
at = 9.8692×10−6 atm ....etc.
Note: mmHg is an abbreviation for millimetres of mercury.
The SI unit for pressure is the pascal (Pa), equal to one
newton per square metre (N·m-2 or kg·m-1·s-2). This special
name for the unit was added in 1971; before that, pressure
in SI was expressed in units such as N/m².
Non-SI measures (still in use in some parts of the world)
include the pound-force per square inch (psi) and the bar.
The cgs unit of pressure is the barye (ba). It is equal to
1 dyn·cm-2.
Pressure is still sometimes expressed in kgf/cm² or grams-
force/cm² (sometimes as kg/cm² and g/cm² without properly
identifying the force units). But using the names kilogram,
gram, kilogram-force, or gram-force (or their symbols) as a
unit of force is expressly forbidden in SI; the unit of
force in SI is the newton (N). The technical atmosphere
(symbol: at) is 1 kgf/cm².
Some meteorologists prefer the hectopascal (hPa) for
atmospheric air pressure, which is equivalent to the older
unit millibar (mbar). Similar pressures are given in
kilopascals (kPa) in practically all other fields, where
the hecto prefix is hardly ever used. In Canadian weather
reports, the normal unit is kPa. The obsolete unit inch of
mercury (inHg, see below) is still sometimes used in the
United States.
The standard atmosphere (atm) is an established constant.
It is approximately equal to typical air pressure at earth
mean sea level and is defined as follows.
standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25
hPa.
Because pressure is commonly measured by its ability to
displace a column of liquid in a manometer, pressures are
often expressed as a depth of a particular fluid (e.g.
inches of water). The most common choices are mercury (Hg)
and water; water is nontoxic and readily available, while
mercury's density allows for a shorter column (and so a
smaller manometer) to measure a given pressure. The press
exerted by a column of liquid of height h and density ρ is
given by the hydrostatic pressure equation as above: p =
hgρ.
Fluid density and local gravity can vary from one reading
to another depending on local factors, so the height of a
fluid column does not define pressure precisely.
When 'millimetres of mercury' or 'inches of mercury' are
quoted today, these units are not based on a physical
column of mercury; rather, they have been given precise
definitions that can be expressed in terms of SI units. The
water-based units still depend on the density of water, a
measured, rather than defined, quantity.
Although no longer favoured in physics, these manometric
units are still encountered in many fields. Blood pressure
is measured in millimetres of mercury in most of the world,
and lung pressures in centimeters of water are still
common. Natural gas pipeline pressures are measured in
inches of water, expressed as '"WC' ('Water Column'). Scuba
divers often use a manometric rule of thumb: the pressure
exerted by ten metres depth of water is approximately equal
to one atmosphere.
Non-SI units presently or formerly in use include the
following:
atmosphere.
manometric units:
centimetre, inch, and millimetre of mercury (Torr).
millimetre, centimetre, metre, inch, and foot of water.
imperial units:
kip, ton-force (short), ton-force (long), pound-force,
ounce-force, and poundal per square inch.
pound-force, ton-force (short), and ton-force (long) per
square foot.
non-SI metric units:
bar, millibar.
kilogram-force, or kilopond, per square centimetre
(technical atmosphere).
gram-force and tonne-force (metric ton-force) per square
centimetre.
barye (dyne per square centimetre).
kilogram-force and tonne-force per square metre.
sthene per square metre (pieze). Stagnation pressure is the
pressure a fluid exerts when it is forced to stop moving.
Consequently, although a fluid moving at higher speed will
have a lower static pressure, it may have a higher
stagnation pressure when forced to a standstill. Static
pressure and stagnation pressure are related by the Mach
number of the fluid. In addition, there can be differences
in pressure due to differences in the elevation (height) of
the fluid. See Bernoulli's equation (note: Bernoulli's
equation only applies for incompressible flow).
The pressure of a moving fluid can be measured using a
Pitot probe, or one of its variations such as a Kiel probe
or Cobra probe, connected to a manometer. Depending on
where the inlet holes are located on the probe, it can
measure static pressure or stagnation pressure.
Now Stress:
Stress (physics)
A mature tree trunk may support a greater force than a fine
steel wire but intuitively we feel that steel is stronger
than wood. To predict the forces which a structure of any
size, made of any material, can sustain without failing, we
need the concept of mechanical stress.
The definition of stress originates in two observations of
the behaviour of a one-dimensional body subject to uniaxial
loading, for example, a steel wire under uniaxial tension:
When the wire is pulled, it stretches. Up to a certain
load, the stretch ratio (current length / initial length)
is proportional to the load divided by the cross-sectional
area of the wire. We therefore define the stress as σ =
F/A.
Failure occurs when the load exceeds a critical value for
the material, the tensile strength multiplied by the cross-
sectional area of the wire, Fc = σt A.
These observations suggest that the quantity that effects
the deformation and failure of materials is stress.
In the more general setting of continuum mechanics, stress
is a measure of the internal distribution of force per unit
area that balances and reacts to the external loads or
boundary conditions applied to a body. Stress is an example
of a type of quantity called a second-order tensor. In 3-
dimensions, a second-order tensor can be represented by a
3x3 square matrix containing nine components. However, in
the absence of body moments, the stress tensor is symmetric
and can be fully specified by six components. These six
components are sometimes written in Voigt notation as a 6x1
column matrix which is often, misleadingly, called the
stress vector.
In N-dimensions, the stress tensor is defined by:
where the are the components of the resultant force vector
acting on a small area which can be represented by a vector
perpendicular to the area element, facing outwards and with
length equal to the area of the element. In elementary
mechanics, the subscripts are often denoted x,y,z rather
than 1,2,3.
In a 1-dimensional system, such as a uniaxially loaded bar,
stress is simply equal to the applied force divided by the
cross-sectional area of the bar (see also pressure). The 2-
D or 3-D cases are more complex. In three dimensions, the
internal force acting on a small area dA of a plane that
passes through a point P can be resolved into three
components: one normal to the plane and two parallel to the
plane (see Figure 1). The normal component divided by dA
gives the normal stress (usually denoted σ), and the
parallel components divided by the area dA give shear
stresses (denoted τ in elementary textbooks). If the area
dA is finite then, strictly, these are average stresses. In
the limit, when dA approaches zero, the stresses become
stresses at the point P. In general, stress varies from
point to point and so is a tensor field.
The components of the stress tensor depend on the
orientation of the plane that passes through the point
under consideration, i.e on the viewpoint of the observer.
This leads to the ridiculous conclusion that the stress on
a structure, and hence its proximity to failure, depends on
the viewpoint of the observer. However, every tensor,
including stress, has invariants that don't depend on the
choice of viewpoint. The length of a first-order tensor,
i.e a vector, is a simple example. The existence of
invariants means that the components seen by one observer
are related, via the tensor transformation relations, to
those seen by any other observer. The transformation
relations for a second-order tensor like stress are
different from those of a first-order tensor, which is why
it is misleading to speak of the stress 'vector'. Mohr's
circle method is a graphical method for performing stress
(or strain) transformations.
Stress can be applied to all types of materials, including
solids, liquids and gases. Static fluids can support normal
or hydrostatic stress (pressure) but will flow under shear
stress. Moving viscous fluids can support shear stress
('dynamic pressure'). Solids can support both shear and
normal stress, even when static. Brittle fracture of solids
is controlled by the maximum principal stress, which is a
normal stress. Ductile fracture of solids is controlled by
the Mises stress, which depends on the shear stresses. Some
materials are however difficult to pigeonhole and their
reaction to stress may be rate-dependent, i.e depend on the
speed with which the stress is applied or temperature
dependent. Toffee, for example, is a brittle solid at room
temperature when the stress rate is high, a ductile solid
when the stress rate is low or the temperature is higher
and a viscous fluid when the temperature is raised even
more.
All real objects occupy 3-dimensional space. However, if
two dimensions are very large or very small compared to the
others, the object may be modelled as one-dimensional. This
simplifies the mathematical modelling of the object. One-
dimensional objects include a piece of wire loaded at the
ends and viewed from the side and a metal sheet loaded on
the face and viewed up close and through the cross-section.
For 1-dimensional objects, the stress tensor has only one
component and is indistinguishable from a scalar. The
simplest definition of stress, σ = F/A, where A is the
initial cross-sectional area prior to the application of
the load, is called engineering stress or nominal stress.
However, when any material is stretched, its cross-
sectional area reduces by an amount that depends on the
Poisson's ratio of the material. Engineering stress
neglects this change in area. The stress axis on a stress-
strain graph is often engineering stress, even though the
sample may undergo a substantial change in cross-sectional
area during testing.
True stress is an alternative definition in which the
initial area is replaced by the current area. In
engineering applications, the initial area is always known
and so calculations using nominal stress are generally
easier. For small deformation, the reduction in cross-
sectional area is small and the distinction between nominal
and true stress is insignificant. This isn't so for the
large deformations typical of elastomers and plastic
materials when the change in cross-sectional areas can be
significant.
In one dimension, true stress is related to nominal stress
via σtrue = σ(1 + ε) where ε is nominal strain and σ is
nominal stress. In uniaxial tension, true stress is then
greater than the nominal stress. The converse holds in
compression.
Example: a steel bolt of diameter 5 mm, has a cross-
sectional area of 19.6 mm2. A load of 50 N induces a stress
(force distributed over the cross section) of σ = 50/19.6 =
2.55 MPa (N/mm2). This can be thought of as each square
millimeter of the bolt supporting 2.55 N of the total load.
In another bolt with half the diameter, and hence a quarter
the cross-sectional area, carrying the same 50 N load, the
stress will be quadrupled (10.2 MPa).
The ultimate tensile strength is a property of a material
and is usually determined experimentally from a uniaxial
tensile test. It allows the calculation of the load that
would cause fracture. The compressive strength is a similar
property for compressive loads. The yield stress is the
value of stress causing plastic deformation.
All real objects occupy 3-dimensional space. However, if
one dimension is very large or very small compared to the
others, the object may be modelled as two-dimensional. This
simplifies the mathematical modelling of the object. Two-
dimensional objects include a piece of wire loaded on the
sides and viewed up close and through the cross-section and
a metal sheet loaded in-plane and viewed face-on.
Notice that the same physical, three-dimensional object can
be modelled as one-dimensional, two-dimensional or even
three-dimensional, depending on the loading and viewpoint
of the observer.
Plane stress is a two-dimensional state of stress (Figure
2). This 2-D state models well the state of stresses in a
flat, thin plate loaded in the plane of the plate. Figure 2
shows the stresses on the x- and y-faces of a differential
element. Not shown in the figure are the stresses in the
opposite faces and the external forces acting on the
material. Since moment equilibrium of the differential
element shows that the shear stresses on the perpendicular
faces are equal, the 2-D state of stresses is characterized
by three independent stress components (σx, σy, τxy). Note
that forces perpendicular to the plane can be abbreviated.
For example, σx is an abbreviation for σxx. This notation
is described further below.
Cauchy was the first to demonstrate that at a given point,
it is always possible to locate two orthogonal planes in
which the shear stress vanishes. These planes are called
the principal planes, while the normal stresses on these
planes are the principal stresses. They are the eigenvalues
of the stress tensor and are orthogonal because the stress
tensor is symmetric (as per the spectral theorem).
Eigenvalues are invariants with respect to choice of basis
and are the roots of the Cayley–Hamilton theorem (although
the term 'the' invariants usually means (I1,I2,I3)). Mohr's
circle is a graphical method of extracting the principal
stresses in a 2-dimensional stress state. The maximum and
minimum principal stresses are the maximum and minimum
possible values of the normal stresses. The maximum
principal stress controls brittle fracture.
Engineers use Mohr's circle to find the planes of maximum
normal and shear stresses, as well as the stresses on known
weak planes. For example, if the material is brittle, the
engineer might use Mohr's circle to find the maximum
component of normal stress (tension or compression); and
for ductile materials, the engineer might look for the
maximum shear stress.
Augustin Louis Cauchy enunciated the principle that, within
a body, the forces that an enclosed volume imposes on the
remainder of the material must be in equilibrium with the
forces upon it from the remainder of the body.
This intuition provides a route to characterizing and
calculating complicated patterns of stress. To be exact,
the stress at a point may be determined by considering a
small element of the body that has an area ΔA, over which a
force ΔF acts. By making the element infinitesimally small,
the stress vector σ is defined as the limit:
Being a tensor, the stress has two directional components:
one for force and one for plane orientation; in three
dimensions these can be two forces within the plane of the
area A, the shear components, and one force perpendicular
to A, the normal component. Therefore the shear stress can
be further decomposed into two orthogonal force components
within the plane. This gives rise to three total stress
components acting on this plane. For example in a plane
orthogonal to the x axis, there can be a normal force
applied in the x direction and a combination of y and z in
plane force components.
The considerations above can be generalized to three
dimensions. However, this is very complicated, since each
shear loading produces shear stresses in one orientation
and normal stresses in other orientations, and vice versa.
Often, only certain components of stress will be important,
depending on the material in question.
The von Mises stress is derived from the distortion energy
theory and is a simple way to combine stresses in three
dimensions to calculate failure criteria of ductile
materials. In this way, the strength of material in a 3-D
state of stress can be compared to a test sample that was
loaded in one dimension.
Because the behavior of a body does not depend on the
coordinates used to measure it, stress can be described by
a tensor. In the absence of body moments, the stress tensor
is symmetric and can always be resolved into the sum of two
symmetric tensors:
a mean or hydrostatic stress tensor, involving only pure
tension and compression; and
a shear or deviatoric stress tensor, involving only shear
stress.
In the case of a fluid, Pascal's law shows that the
hydrostatic stress is the same in all directions, at least
to a first approximation, so can be captured by the scalar
quantity pressure. Thus, in the case of a solid, the
hydrostatic (or isostatic) pressure p is defined as one
third of the trace of the tensor, i.e., the mean of the
diagonal terms.
In the generalized stress tensor notation, the tensor
components are written σij, where i and j are in {1;2;3}.
(caution: subscript notation in this section is different
from the rest of the article - the order of subscripts is
reversed)
The first step is to number the sides of the cube. When the
lines are parallel to a vector base , then:
the sides perpendicular to are called j and -j; and
from the center of the cube, points toward the j side,
while the -j side is at the opposite.
σij is then the component along the i axis that applies on
the j side of the cube. (Or in books in the English
language, σij is the stress on the i face acting in the j
direction -- the transpose of the subscript notation
herein. But transposing the subscript notation produces the
same stress tensor, since a symmetric matrix is equal to
its transpose.)
This generalized notation allows an easy writing of
equations of the continuum mechanics, such as the
generalized Hooke's law:
The correspondence with the former notation is thus:
x → 1
y → 2
z → 3
σxx → σ11
τxy → σ12
τxz → σ13
...
The fact that the Newtonian stress is a symmetric tensor
follows from some simple considerations. The force on a
small volume element will be the sum of all the stress
forces over the surface area of that element. Suppose we
have a volume element in the form of a long bar with a
triangular cross section, where the triangle is a right
triangle. We can neglect the forces on the ends of the bar,
because they are small compared to the faces of the bar.
Let be the vector area of one face of the bar, be the area
of the other, and be the area of the "hypotenuse face" of
the bar. It can be seen that
Let's say is the force on area and likewise for the other
faces. Since the stress is by definition the force per unit
area, it is clear that
The total force on the volume element will be:
Let's suppose that the volume element contains mass, at a
constant density. The important point is that if we make
the volume smaller, say by halving all lengths, the area
will decrease by a factor of four, while the volume will
decrease by a factor of eight. As the size of the volume
element goes to zero, the ratio of area to volume will
become infinite. The total stress force on the element is
proportional to its area, and so as the volume of the
element goes to zero, the force/mass (i.e. acceleration)
will also become infinite, unless the total force is zero.
In other words:
This, along with the second equation above, proves that the
σ function is a linear vector operator (i.e. a tensor). By
an entirely analogous argument, we can show that the total
torque on the volume element (due to stress forces) must be
zero, and that it follows from this restriction that the
stress tensor must be symmetric.
However, there are two fundamental ways in which this mode
of thinking can be misleading. First, when applying this
argument in tandem with the underlying assumption from
continuum mechanics that the Knudsen number is strictly
less than one, then in the limit , the symmetry assumptions
in the stress tensor may break down. This is the case of
Non-Newtonian fluid, and can lead to rotationally non-
invariant fluids, such as polymers. The other case is when
the system is operating on a purely finite scale, such as
is the case in mechanics where Finite deformation tensors
are used.
The state of stress as defined by the stress tensor is an
equilibrium state if the following conditions are satisfied:
σij are the components of the tensor, and f 1 , f 2 , and f
3 are the body forces (force per unit volume).
These equations can be compactly written using Einstein
notation in which repeated indices are summed. Defining as
the equilibrium conditions are written:
The equilibrium conditions may be derived from the
condition that the net force on an infinitesimal volume
element must be zero. Consider an infinitesimal cube
aligned with the x1, x2, and x3 axes, with one corner at xi
and the opposite corner at xi + dxi and having each face of
area dA. Consider just the faces of the cube which are
perpendicular to the x1 axis. The area vector for the near
face is [ − dA,0,0] and for the far face it is [dA,0,0].
The net stress force on these two opposite faces is
A similar calculation can be carried out for the other
pairs of faces. The sum of all the stress forces on the
infinitesimal cube will then be
Since the net force on the cube must be zero, it follows
that this stress force must be balanced by the force per
unit volume fi on the cube (e.g., due to gravitation,
electromagnetic forces, etc.) which yields the equilibrium
conditions.
Equilibrium also requires that the resultant moment on the
cube of material must be zero. Taking the moment of the
forces above about any suitable point, it follows that, for
equilibrium in the absence of body moments
σij = σji.
The stress tensor is then symmetric and the subscripts can
be written in either order.
As with force, stress cannot be measured directly but is
usually inferred from measurements of strain and knowledge
of elastic properties of the material. Devices capable of
measuring stress indirectly in this way are strain gages
and piezoresistors.
The SI unit for stress is the pascal (symbol Pa), the same
as that of pressure. Since the pascal is so small,
engineering quantities are usually measured in megapascals
(MPa) or gigapascals (GPa) In US Customary units, stress is
expressed in pounds-force per square inch (psi). See also
pressure.
Residual stresses are stresses that remain after the
original cause of the stresses has been removed. Residual
stresses occur for a variety of reasons, including
inelastic deformations and heat treatment. Heat from
welding may cause localized expansion. When the finished
weldment cools, some areas cool and contract more than
others, leaving residual stresses. Castings may also have
large residual stresses due to uneven cooling.
While uncontrolled residual stresses are undesirable, many
designs rely on them. For example, toughened glass and
prestressed concrete rely on residual stress to prevent
brittle failure. Similarly, a gradient in martensite
formation leaves residual stress in some swords with
particularly hard edges (notably the katana), which can
prevent the opening of edge cracks. In certain types of gun
barrels made with two telescoping tubes forced together,
the inner tube is compressed while the outer tube
stretches, preventing cracks from opening in the rifling
when the gun is fired. These tubes are often heated or
dunked in liquid nitrogen to aid assembly.
Press fits are the most common intentional use of residual
stress. Automotive wheel studs, for example, are pressed
into holes on the wheel hub. The holes are smaller than the
studs, requiring force to drive the studs into place. The
residual stresses fasten the parts together. Nails are
another example.
|
| Ishfaq
|
To Refer this Site to Your Friends 
Site Map
