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Stiffness

For pain and/or loss of range of motion of a joint, see joint stiffness. For the term regarding the stability of a differential equation, see stiff equation.

Stiffness is the extent to which an object resists deformation in response to an applied force.

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

The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.

Contents

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 single degree of freedom (DOF) (for example, stretching or compression of a rod), the stiffness is defined as

k = F δ {\displaystyle k={\frac {F}{\delta }}}
where,
  • F {\displaystyle F} is the force on the body
  • δ {\displaystyle \delta } 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 ( N / m {\displaystyle N/m} ). 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 M {\displaystyle M} degrees of freedom a M × M {\displaystyle 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 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 describes all possible stretch and shear parameters is given by the elasticity tensor.

"Flexibility" redirects here. For other uses, see Flexibility (disambiguation).

The inverse of stiffness is flexibility or compliance, 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.

Twist, by angle α {\displaystyle \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 a , {\displaystyle ka,} given by

k = M θ {\displaystyle k={\frac {M}{\theta }}}
where
  • M {\displaystyle M} is the applied moment
  • θ {\displaystyle \theta } 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

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 and its shape and boundary conditions. For example, for an element in tension or compression, the axial stiffness is

k = E A L {\displaystyle k=E\cdot {\frac {A}{L}}}
where

Similarly, the torsional stiffness of a straight section is

k = G J L {\displaystyle k=G\cdot {\frac {J}{L}}}
where

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 modulus can be thought of as a measure of the stiffness of a structure.

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 is undesirable, while a low modulus of elasticity is required when flexibility is needed.

In biology, the stiffness of the extracellular matrix is important for guiding the migration of cells in a phenomenon called durotaxis.

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.[citation needed] This is of significance to patients with traumatic injuries to the skin, whereby the liability 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.

  1. Baumgart F. (2000). "Stiffness--an unknown world of mechanical science?". Injury. Elsevier. 31: 14–84. doi:10.1016/S0020-1383(00)80040-6. “Stiffness” = “Load” divided by “Deformation”
  2. Martin Wenham (2001), "Stiffness and flexibility", 200 science investigations for young students, p. 126, ISBN 978-0-7619-6349-3
  3. 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.
  4. Chattopadhyay, S.; Raines, R. (August 2014). "Collagen-Based Biomaterials for Wound Healing". Biopolymers. 101 (8): 821–833. doi:10.1002/bip.22486. PMC4203321. PMID 24633807.
  5. 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.

Stiffness
Stiffness Language Watch Edit For pain and or loss of range of motion of a joint see joint stiffness For the term regarding the stability of a differential equation see stiff equation Stiffness is the extent to which an object resists deformation in response to an applied force 1 Extension of a coil spring d displaystyle delta caused by an axial force F displaystyle F The complementary concept is flexibility or pliability the more flexible an object is the less stiff it is 2 Contents 1 Calculations 2 Compliance 3 Rotational stiffness 4 Relationship to elasticity 5 Applications 6 See also 7 ReferencesCalculations EditThe 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 single degree of freedom DOF for example stretching or compression of a rod the stiffness is defined ask F d displaystyle k frac F delta where F displaystyle F is the force on the body d displaystyle delta 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 N m displaystyle N m 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 M displaystyle M degrees of freedom a M M displaystyle 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 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 describes all possible stretch and shear parameters is given by the elasticity tensor Compliance Edit Flexibility redirects here For other uses see Flexibility disambiguation The inverse of stiffness is flexibility or compliance typically measured in units of metres per newton In rheology it may be defined as the ratio of strain to stress 3 and so take the units of reciprocal stress for example 1 Pa Rotational stiffness Edit Twist by angle a displaystyle 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 a displaystyle ka given byk M 8 displaystyle k frac M theta where M displaystyle M is the applied moment 8 displaystyle theta 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 twistRelationship to elasticity EditThe 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 and its shape and boundary conditions For example for an element in tension or compression the axial stiffness isk E A L displaystyle k E cdot frac A L where E displaystyle E is the tensile elastic modulus or Young s modulus A displaystyle A is the cross sectional area L displaystyle L is the length of the element Similarly the torsional stiffness of a straight section isk G J L displaystyle k G cdot frac J L where G displaystyle G is the rigidity modulus of the material J displaystyle J is the 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 Young s modulus can be thought of as a measure of the stiffness of a structure Applications EditThe 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 is undesirable while a low modulus of elasticity is required when flexibility is needed In biology the stiffness of the extracellular matrix is important for guiding the migration of cells in a phenomenon called durotaxis 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 4 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 citation needed This is of significance to patients with traumatic injuries to the skin whereby the liability 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 5 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 EditBending stiffness Compliant mechanism Elasticity physics Physical property when materials or objects return to original shape after deformation Elastic modulus Physical property that measures stiffness of material Elastography Hardness Resistance to localized plastic deformation from mechanical indentation or abrasion Hooke s law Principle of physics that states that the force F needed to extend or compress a spring by some distance X scales linearly with respect to that distance Mechanical impedance Moment of inertia Scalar measure of the rotational inertia with respect to a fixed axis of rotation Shore durometer Spring device Elastic object that stores mechanical energy Stiffness mathematics Stiffness tensor Young s modulus Mechanical property that measures stiffness of a solid materialReferences Edit Baumgart F 2000 Stiffness an unknown world of mechanical science Injury Elsevier 31 14 84 doi 10 1016 S0020 1383 00 80040 6 Stiffness Load divided by Deformation Martin Wenham 2001 Stiffness and flexibility 200 science investigations for young students p 126 ISBN 978 0 7619 6349 3 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 Nedelec Bernadette Correa Jose 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 Retrieved from https en wikipedia org w index php title Stiffness amp oldid 1050519918, wikipedia, wiki, book,

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