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Stiffness

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From Wikipedia, the free encyclopedia

Resistance to deformation in response to force

For other uses, seeStiff (disambiguation).

Extension of a coil spring, $\delta ,$ caused by an axial force, $F . {\displaystyle F.}$

{\displaystyle \delta ,} — Extension of a coil spring, $\delta ,$ caused by an axial force, $F . {\displaystyle F.}$

Stiffness is the extent to which an object resistsdeformation in response to an appliedforce.^[1]

Theinverse of stiffness isflexibility or pliability: the more flexible an object is, the less stiff it is.^[2] Other terms with similar meanings are compliance and elasticity.

Calculations

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The stiffness, $k, {\displaystyle k,}$ of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a singledegree of freedom (DOF) (for example, stretching or compression of a rod), the stiffness is defined as $k={\frac {F}{\delta }}$ where,

$F {\displaystyle F}$ is the force on the body
$\delta$ is thedisplacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)

Stiffness is usually defined underquasi-static conditions, but sometimes under dynamic loading.^[3]

In theInternational System of Units, stiffness is typically measured innewtons per meter ( $N/m$ ). In Imperial units, stiffness is typically measured inpounds (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 horizontalbeam can undergo both a verticaldisplacement and a rotation relative to its undeformed axis. When there are $M {\displaystyle M}$ degrees of freedom a $M\times M$ 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 terminfluence 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.

Theelasticity 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

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Theinverse of stiffness isflexibility, elasticity orcompliance, typically measured in units of metres per newton. Inrheology, it may be defined as the ratio ofstrain tostress,^[4] and so take the units of reciprocal stress, for example, 1/Pa.

Rotational stiffness

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Twist, by angle $\alpha$ of a cylindrical bar, with length $L, {\displaystyle L,}$ caused by an axial moment, $M . {\displaystyle M.}$

{\displaystyle \alpha } — Twist, by angle $\alpha$ of a cylindrical bar, with length $L, {\displaystyle L,}$ caused by an axial moment, $M . {\displaystyle M.}$

A body may also have a rotational stiffness, $k, {\displaystyle k,}$ given by $k={\frac {M}{\theta }}$ where

$M {\displaystyle M}$ is the appliedmoment
$\theta$ is the rotation angle

In the SI system, rotational stiffness is typically measured innewton-metres perradian.

In the SAE system, rotational stiffness is typically measured in inch-pounds perdegree.

Further measures of stiffness are derived on a similar basis, including:

shear stiffness - the ratio of appliedshear force to shear deformation
torsional stiffness - the ratio of appliedtorsion moment to the angle of twist

Relationship to elasticity

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Theelastic 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 anintensive property of the material; stiffness, on the other hand, is anextensive property of the solid body that is dependent on the materialand its shape and boundary conditions. For example, for an element intension orcompression, the axial stiffness is $k=E\cdot {\frac {A}{L}}$ where

$E {\displaystyle E}$ is the (tensile) elastic modulus (orYoung's modulus),
$A {\displaystyle A}$ is thecross-sectional area,
$L {\displaystyle L}$ is thelength of the element.

Similarly, the torsional stiffness of a straight section is $k=G\cdot {\frac {J}{L}}$ where

$G {\displaystyle G}$ is therigidity modulus of the material,
$J {\displaystyle J}$ is thetorsion 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,Young's moduluscan be thought of as a measure of the stiffness of a structure.

Applications

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The stiffness of a structure is of principal importance in many engineering applications, so themodulus of elasticity is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought whendeflection is undesirable, while a low modulus of elasticity is required when flexibility is needed.

In biology, the stiffness of theextracellular matrix is important for guiding the migration of cells in a phenomenon calleddurotaxis.

Another application of stiffness finds itself inskin biology. The skin maintains its structure due to its intrinsic tension, contributed to bycollagen, an extracellular protein that accounts for approximately 75% of its dry weight.^[5] 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.^[6] 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 pathologicalscar. 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,^[7] and the method has been applied within clinical and industrial settings to monitor both pathophysiological sequelae, and the effects of treatments on skin.

References

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^Baumgart F. (2000). "Stiffness--an unknown world of mechanical science?".Injury.31. Elsevier:14–84.doi:10.1016/S0020-1383(00)80040-6."Stiffness" = "Stress" divided by "strain"
^Martin Wenham (2001), "Stiffness and flexibility",200 science investigations for young students, SAGE Publications, p. 126,ISBN 978-0-7619-6349-3
^Escudier, Marcel; Atkins, Tony (2019).A Dictionary of Mechanical Engineering (2 ed.). Oxford University Press.doi:10.1093/acref/9780198832102.001.0001.ISBN 978-0-19-883210-2.
^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.
^Chattopadhyay, S.; Raines, R. (August 2014)."Collagen-Based Biomaterials for Wound Healing".Biopolymers.101 (8):821–833.doi:10.1002/bip.22486.PMC 4203321.PMID 24633807.
^Graham, Helen K; McConnell, James C; Limbert, Georges; Sherratt, Michael J (February 2019)."How stiff is skin?".Experimental Dermatology.28:4–9.doi:10.1111/exd.13826.PMID 30698873.
^Nedelec, Bernadette; Correa, José; de Oliveira, Ana; LaSalle, Leo; Perrault, Isabelle (2014). "Longitudinal burn scar quantification".Burns.40 (8):1504–1512.doi:10.1016/j.burns.2014.03.002.PMID 24703337.