![Stiffness of a coil spring](https://upload.wikimedia.org/wikipedia/commons/6/6d/Stiffness_of_a_coil_spring.png)
Stiffness is the extent to which an object resists
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
in response to an applied
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness,
of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single
degree of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
(DOF) (for example, stretching or compression of a rod), the stiffness is defined as
where,
*
is the force on the body
*
is the
displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)
In the
International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
, stiffness is typically measured in
newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
s per meter (
). In Imperial units, stiffness is typically measured in
pounds (lbs) per inch.
Generally speaking,
deflections (or motions) of an infinitesimal element (which is viewed as a point) in an elastic body can occur along multiple DOF (maximum of six DOF at a point). For example, a point on a horizontal
beam
Beam may refer to:
Streams of particles or energy
*Light beam, or beam of light, a directional projection of light energy
**Laser beam
*Particle beam, a stream of charged or neutral particles
**Charged particle beam, a spatially localized grou ...
can undergo both a vertical
displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
and a rotation relative to its undeformed axis. When there are
degrees of freedom a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
must be used to describe the stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or different points) or the same degree of freedom at two different points. In industry, the term influence coefficient is sometimes used to refer to the coupling stiffness.
It is noted that for a body with multiple DOF, the equation above generally does not apply since the applied force generates not only the deflection along its direction (or degree of freedom) but also those along with other directions.
For a body with multiple DOF, to calculate a particular direct-related stiffness (the diagonal terms), the corresponding DOF is left free while the remaining should be constrained. Under such a condition, the above equation can obtain the direct-related stiffness for the degree of unconstrained freedom. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses.
The
elasticity tensor
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
describes all possible stretch and shear parameters is given by the
elasticity tensor
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
.
Compliance
The
inverse of stiffness is or , typically measured in units of metres per newton. In
rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid ( liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an appl ...
, it may be defined as the ratio of strain to stress, and so take the units of reciprocal stress, for example, 1/
Pa.
Rotational stiffness
![Angle torsion cylindre](https://upload.wikimedia.org/wikipedia/commons/f/fb/Angle_torsion_cylindre.svg)
A body may also have a rotational stiffness,
given by
where
*
is the applied
moment
*
is the rotation
In the SI system, rotational stiffness is typically measured in
newton-metre
The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...
s per
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
.
In the SAE system, rotational stiffness is typically measured in inch-
pounds per
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
.
Further measures of stiffness are derived on a similar basis, including:
* shear stiffness - the ratio of applied
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
force to shear deformation
* torsional stiffness - the ratio of applied
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
moment to the angle of twist
Relationship to elasticity
The
elastic modulus
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
of a material is not the same as the stiffness of a component made from that material. Elastic modulus is a property of the constituent material; stiffness is a property of a structure or component of a structure, and hence it is dependent upon various physical dimensions that describe that component. That is, the modulus is an
intensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is on ...
of the material; stiffness, on the other hand, is an
extensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
of the solid body that is dependent on the material its shape and boundary conditions. For example, for an element in
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
or
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
, the axial stiffness is
where
*
is the (tensile) elastic modulus (or
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
),
*
is the
cross-sectional area,
*
is the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of the element.
Similarly, the torsional stiffness of a straight section is
where
*
is the
rigidity modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
of the material,
*
is the
torsion constant The torsion constant is a geometrical property of a bar's cross-section which is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear-elastic bar. The torsion constant, together w ...
for the section.
Note that the torsional stiffness has dimensions
orce
Orce is a municipality located in the province of Granada, in southeastern Spain. According to the 2009 census (INE), the city has a population of 1333 inhabitants.
Paleoanthropology
Orce is the location of the paleo- archaeological sites known ...
*
ength/
ngle so that its SI units are N*m/rad.
For the special case of unconstrained uniaxial tension or compression,
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
be thought of as a measure of the stiffness of a structure.
Applications
The stiffness of a structure is of principal importance in many engineering applications, so the
modulus of elasticity
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought when
deflection
Deflection or deflexion may refer to:
Board games
* Deflection (chess), a tactic that forces an opposing chess piece to leave a square
* Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers
Mechanics
* Deflection ...
is undesirable, while a low modulus of elasticity is required when flexibility is needed.
In biology, the stiffness of the
extracellular matrix
In biology, the extracellular matrix (ECM), also called intercellular matrix, is a three-dimensional network consisting of extracellular macromolecules and minerals, such as collagen, enzymes, glycoproteins and hydroxyapatite that provide stru ...
is important for guiding the migration of cells in a phenomenon called
durotaxis
Durotaxis is a form of cell migration in which cells are guided by rigidity gradients, which arise from differential structural properties of the extracellular matrix (ECM). Most normal cells migrate up rigidity gradients (in the direction of great ...
.
Another application of stiffness finds itself in
skin
Skin is the layer of usually soft, flexible outer tissue covering the body of a vertebrate animal, with three main functions: protection, regulation, and sensation.
Other cuticle, animal coverings, such as the arthropod exoskeleton, have diffe ...
biology. The skin maintains its structure due to its intrinsic tension, contributed to by
collagen
Collagen () is the main structural protein in the extracellular matrix found in the body's various connective tissues. As the main component of connective tissue, it is the most abundant protein in mammals, making up from 25% to 35% of the whole ...
, an extracellular protein that accounts for approximately 75% of its dry weight. The pliability of skin is a parameter of interest that represents its firmness and extensibility, encompassing characteristics such as elasticity, stiffness, and adherence. These factors are of functional significance to patients. This is of significance to patients with traumatic injuries to the skin, whereby the pliability can be reduced due to the formation and replacement of healthy skin tissue by a pathological
scar
A scar (or scar tissue) is an area of fibrous tissue that replaces normal skin after an injury. Scars result from the biological process of wound repair in the skin, as well as in other organs, and tissues of the body. Thus, scarring is a na ...
. This can be evaluated both subjectively, or objectively using a device such as the Cutometer. The Cutometer applies a vacuum to the skin and measures the extent to which it can be vertically distended. These measurements are able to distinguish between healthy skin, normal scarring, and pathological scarring,
and the method has been applied within clinical and industrial settings to monitor both pathophysiological sequelae, and the effects of treatments on skin.
See also
References
{{Authority control
Physical quantities
Continuum mechanics
Structural analysis