Stiffness is the extent to which an object resists
deformation in response to an applied
force.
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 (DOF) (for example, stretching or compression of a rod), the stiffness is defined as
where,
*
is the force on the body
*
is the
displacement 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, stiffness is typically measured in
newtons 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 can undergo both a vertical
displacement 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'' (franchi ...
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
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
of stiffness is or , typically measured in units of metres per newton. In
rheology, 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
A body may also have a rotational stiffness,
given by
where
*
is the applied
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
*
is the rotation
In the SI system, rotational stiffness is typically measured in
newton-metres per
radian.
In the SAE system, rotational stiffness is typically measured in inch-
pounds per
degree.
Further measures of stiffness are derived on a similar basis, including:
* shear stiffness - the ratio of applied
shear force to shear deformation
* torsional stiffness - the ratio of applied
torsion moment to the angle of twist
Relationship to elasticity
The
elastic modulus 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 of the material; stiffness, on the other hand, is an
extensive property of the solid body that is dependent on the material its shape and boundary conditions. For example, for an element in
tension or
compression, the axial stiffness is
where
*
is the (tensile) elastic modulus (or
Young's modulus),
*
is the
cross-sectional area
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The ...
,
*
is the
length of the element.
Similarly, the torsional stiffness of a straight section is
where
*
is the
rigidity modulus of the material,
*
is the
torsion constant 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 know ...
*
ength/
ngle so that its SI units are N*m/rad.
For the special case of unconstrained uniaxial tension or compression,
Young's modulus 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 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 s ...
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 gre ...
.
Another application of stiffness finds itself in
skin biology. The skin maintains its structure due to its intrinsic tension, contributed to by
collagen, 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. 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