Stiffness
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2008-07-15T09:40:23Z
128.250.80.15
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{{for|pain and/or loss of range of motion of a joint|joint stiffness}}
'''Stiffness''' is the resistance of an [[Elasticity (physics)|elastic]] body to [[deflection]] or [[deformation]] by an applied [[force]]. It is an [[intensive and extensive properties|extensive material property]].
==Definition==
The stiffness, ''k'', of a body is
:<math>k=\frac {P} {\delta} </math>
where
:''P'' is the applied force
:δ is the deflected distance
In the [[International System of Units]], stiffness is typically measured in [[newton]]s per [[metre]].
As both the applied force and [[deflection#Construction|deflection]] are [[vector (spatial)|vector]]s (respectively '''''P''''' and '''δ'''), in general their relationship is characterised by a stiffness matrix, '''''k''''', where:
:<math> P=k \delta \,</math>
The deflection can, in general, refer to a point distinct from that where the force is applied and a complicated structure will not deflect purely in the same direction as an applied force. The stiffness matrix enables such systems to be characterised in straightforward terms.
The [[inverse]] of stiffness is ''[[compliance]]'', typically measured in units of metres per newton. In 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, ''e.g''. 1/[[pascal (unit)|Pa]].
== Rotational stiffness ==
A body may also have a rotational stiffness, ''k'', given by
:<math>k=\frac {M} {\theta} </math>
where
: ''M'' is the applied [[moment (physics)|moment]]
: ''θ'' is the rotation
In the SI system, rotational stiffness is typically measured in [[newton-metre]]s per [[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 - ratio of applied [[shear stress|shear]] force to shear deformation
* torsional stiffness - ratio of applied [[torsion (mechanics)|torsion]] moment to angle of twist
== Relationship to elasticity ==
In general, [[elastic modulus]] is not the same as stiffness. Elastic modulus is a property of the constituent material; stiffness is a property of a solid body. That is, the modulus is an [[intensive and extensive properties|intensive property]] of the material; stiffness, on the other hand, is an [[intensive and extensive properties|extensive property]] of the solid body dependent on the material ''and'' the shape and boundary conditions. For example, for an element in [[tension (mechanics)|tension]] or [[physical compression|compression]], the axial stiffness is
:<math>k=\frac {AE} {L} </math>
where
:''A'' is the cross-sectional area,
:''E'' is the [[Young's modulus]],
:''L'' is the length of the element.
For the special case of unconstrained uniaxial tension or compression, [[Young's modulus|Young's modulus of elasticity]] ''can'' be thought of as a measure of the stiffness of a material.
== Use in engineering ==
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|deflections]] are undesirable, while a low [[modulus of elasticity]] is required when flexibility is needed.
== References ==
{{Reflist}}
== See also ==
*[[Elasticity (physics)|Elasticity]]
*[[Elastic modulus]]
*[[Hardness]]
*[[Moment of inertia]]
*[[Young's modulus]]
*[[Hooke's law]]
[[Category:Physical quantity]]
[[Category:Classical mechanics]]
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