View source for Physics:Stiffness

{{Short description|Resistance to deformation in response to force}}
[[File:Stiffness of a coil spring.png|thumb|right|Extension of a coil spring, <math>\delta,</math> caused by an axial force, <math>F.</math>]]
'''Stiffness''' is the extent to which an object resists [[Physics:Deformation (mechanics)|deformation]] in response to an applied [[Physics:Force|force]].<ref>{{cite journal
| title = Stiffness--an unknown world of mechanical science?
| journal = Injury
| author = Baumgart F.
| year = 2000
| volume = 31
| pages = 14–84
| publisher = Elsevier
| quote = “Stiffness” = “Stress” divided by “strain”
| doi=10.1016/S0020-1383(00)80040-6}}</ref>

The complementary concept is '''flexibility''' or pliability: the more flexible an object is, the less stiff it is.<ref>{{citation |page=126 |chapter=Stiffness and flexibility |title=200 science investigations for young students |author=Martin Wenham |year=2001 |publisher=SAGE Publications |isbn=978-0-7619-6349-3}}</ref>

==Calculations==
The stiffness, <math>k,</math> of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single [[Physics:Degrees of freedom (mechanics)|degree of freedom]] (DOF) (for example, stretching or compression of a rod), the stiffness is defined as
<math display=block>k = \frac {F}{\delta}</math>
where,
* <math>F</math> is the force on the body
* <math>\delta</math> is the [[Displacement (vector)|displacement]] produced by the force along the same degree of freedom (for instance,  the change in length of a stretched spring)
Stiffness is usually defined under [[Physics:Quasistatic loading|quasi-static conditions]], but sometimes under [[Dynamic loading|dynamic loading]].<ref>{{Cite book |last1=Escudier |first1=Marcel |url=http://www.oxfordreference.com/view/10.1093/acref/9780198832102.001.0001/acref-9780198832102 |title=A Dictionary of Mechanical Engineering |last2=Atkins |first2=Tony |date=2019 |publisher=Oxford University Press |isbn=978-0-19-883210-2 |edition=2 |language=en |doi=10.1093/acref/9780198832102.001.0001}}</ref>

In the [[International System of Units]], stiffness is typically measured in [[Newton (unit)|newton]]s per meter (<math>N/m</math>). In Imperial units, stiffness is typically measured in [[Pound (force)|pound]]s (lbs) per inch.

Generally speaking, [[Engineering:Deflection (engineering)|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 (vector)|displacement]] and a rotation relative to its undeformed axis. When there are <math>M</math> degrees of freedom a <math>M \times M</math> [[Matrix (mathematics)|matrix]] 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 [[Physics:Elasticity tensor|elasticity tensor]] is a generalization that describes all possible stretch and shear parameters.

A single spring may intentionally be designed to have variable (non-linear) stiffness throughout its displacement.

== Compliance ==
The [[Multiplicative inverse|inverse]] of stiffness is {{em|flexibility}} or {{em|compliance}}, typically measured in units of metres per newton. In [[Physics:Rheology|rheology]], it may be defined as the ratio of strain to stress,<ref>V. GOPALAKRISHNAN and CHARLES F. ZUKOSKI;  "Delayed flow in thermo-reversible colloidal gels";  Journal of Rheology;  Society of Rheology, U.S.A.;  July/August 2007;  51 (4): pp. 623–644.</ref> and so take the units of reciprocal stress, for example, 1/[[Pascal (unit)|Pa]].

==Rotational stiffness==<!-- Torsional rigidity redirects here -->
[[File:Angle torsion cylindre.svg|thumb|right|Twist, by angle <math>\alpha</math> of a cylindrical bar, with length <math>L,</math> caused by an axial moment, <math>M.</math>]]

A body may also have a rotational stiffness, <math>k,</math> given by
<math display=block>k = \frac{M}{\theta}</math>
where
* <math>M</math> is the applied [[Physics:Moment|moment]]
* <math>\theta</math> is the rotation angle

In the SI system, rotational stiffness is typically measured in [[Newton-metre|newton-metre]]s per [[Radian|radian]].

In the SAE system, rotational stiffness is typically measured in inch-[[Pound (force)|pound]]s per [[Degree (angle)|degree]].

Further measures of stiffness are derived on a similar basis, including:
* shear stiffness - the ratio of applied [[Physics:Shear stress|shear]] force to shear deformation
* torsional stiffness - the ratio of applied [[Physics:Torsion (mechanics)|torsion]] moment to the angle of twist

== Relationship to elasticity ==
The [[Physics:Elastic modulus|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 [[Physics:Intensive and extensive properties|intensive property]] of the material; stiffness, on the other hand, is an [[Physics:Intensive and extensive properties|extensive property]] of the solid body that is dependent on the material {{em|and}} its shape and boundary conditions.  For example, for an element in tension or compression, the axial stiffness is
<math display=block>k = E \cdot \frac{A}{L}</math>
where
* <math>E</math> is the (tensile) elastic modulus (or [[Physics:Young's modulus|Young's modulus]]),
* <math>A</math> is the [[Cross section (geometry)#Area and volume|cross-sectional area]],
* <math>L</math> is the [[Physics:Length|length]] of the element.

Similarly, the torsional stiffness of a straight section is
<math display=block>k = G \cdot \frac{J}{L}</math>
where
* <math>G</math> is the rigidity modulus of the material,
* <math>J</math> is the [[Torsion constant|torsion constant]] for the section.

Note that the torsional stiffness has dimensions [force] * [length] / [angle], so that its SI units are N*m/rad.

For the special case of unconstrained uniaxial tension or compression, [[Physics:Young's modulus|Young's modulus]] {{em|can}} 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 [[Engineering:Deflection (engineering)|deflection]] is undesirable, while a low modulus of elasticity is required when flexibility is needed.

In biology, the stiffness of the [[Biology:Extracellular matrix|extracellular matrix]] is important for guiding the migration of cells in a phenomenon called [[Philosophy:Durotaxis|durotaxis]].

Another application of stiffness finds itself in [[Biology:Skin|skin]] biology. The skin maintains its structure due to its intrinsic tension, contributed to by [[Biology:Collagen|collagen]], an extracellular protein that accounts for approximately 75% of its dry weight.<ref>{{cite journal|last1=Chattopadhyay|first1=S.|last2=Raines|first2=R.|title=Collagen-Based Biomaterials for Wound Healing|journal=Biopolymers|date=August 2014|volume=101|issue=8|pages=821–833|doi= 10.1002/bip.22486|pmid=24633807|pmc=4203321}}</ref> 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.<ref>{{cite journal |last1=Graham |first1=Helen K  |last2=McConnell |first2=James C |last3=Limbert |first3= Georges|last4=Sherratt |first4= Michael J |date=February 2019 |title=How stiff is skin? |url= |journal=Experimental dermatology |volume=28 |issue= |pages=4-9 |doi=10.1111/exd.13826 |access-date=|doi-access=free }}</ref> 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,<ref>{{cite journal|last1=Nedelec|first1=Bernadette|last2=Correa|first2=José|last3=de Oliveira|first3=Ana|last4=LaSalle|first4=Leo|last5=Perrault|first5=Isabelle|title=Longitudinal burn scar quantification|journal=Burns|volume=40|issue=8|pages=1504–1512|date=2014|doi= 10.1016/j.burns.2014.03.002|pmid=24703337}}</ref> 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 ==

{{Columns-list|colwidth=30em|
* {{annotated link|Bending stiffness}}
* {{annotated link|Engineering:Compliant mechanism}}
* {{annotated link|Physics:Elasticity}}
* {{annotated link|Physics:Elastic modulus}}
* {{annotated link|Medicine:Elastography}}
* {{annotated link|Physics:Hardness}}
* {{annotated link|Physics:Hooke's law}}
* {{annotated link|Physics:Mechanical impedance}}
* {{annotated link|Physics:Moment of inertia}}
* {{annotated link|Shore durometer}}
* {{annotated link|Engineering:Spring (device)}}
* {{annotated link|Physics:Young's modulus}}
}}

== References ==

{{reflist}}


[[Category:Physical quantities]]
[[Category:Continuum mechanics]]
[[Category:Structural analysis]]

{{Sourceattribution|Stiffness}}