Stiffness

Last updated
Extension of a coil spring,
d
,
{\displaystyle \delta ,}
caused by an axial force,
F
.
{\displaystyle F.} Stiffness of a coil spring.png
Extension of a coil spring, caused by an axial force,

Stiffness is the extent to which an object resists deformation in response to an applied force. [1]

Contents

The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. [2]

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,

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

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 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 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 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, [4] and so take the units of reciprocal stress, for example, 1/Pa.

Rotational stiffness

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

A body may also have a rotational stiffness, given by

where

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:

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

where

Similarly, the torsional stiffness of a straight section is

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.

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 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. [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 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, [7] 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

Related Research Articles

<span class="mw-page-title-main">Speed of sound</span> Speed of sound wave through elastic medium

The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At 20 °C (68 °F), the speed of sound in air is about 343 m/s, or 1 km in 2.91 s or one mile in 4.69 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating. At 0 °C (32 °F), the speed of sound in air is about 331 m/s. More simply, the speed of sound is how fast vibrations travel.

<span class="mw-page-title-main">Young's modulus</span> Mechanical property that measures stiffness of a solid material

Young's modulus is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress applied to the object and the resulting axial strain in the linear elastic region of the material.

<span class="mw-page-title-main">Hooke's law</span> Physical law: force needed to deform a spring scales linearly with distance

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, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.

In engineering, deformation refers to the change in size or shape of an object.

<span class="mw-page-title-main">Poisson's ratio</span> Measure of material deformation perpendicular to loading

In materials science and solid mechanics, Poisson's ratioν (nu) is a measure of the Poisson effect, the deformation of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, ν is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3. The ratio is named after the French mathematician and physicist Siméon Poisson.

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.

The field of strength of materials typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failure modes takes into account the properties of the materials such as its yield strength, ultimate strength, Young's modulus, and Poisson's ratio. In addition, the mechanical element's macroscopic properties such as its length, width, thickness, boundary constraints and abrupt changes in geometry such as holes are considered.

<span class="mw-page-title-main">Spring (device)</span> Elastic object that stores mechanical energy

A spring is a device consisting of an elastic but largely rigid material bent or molded into a form that can return into shape after being compressed or extended. Springs can store energy when compressed. In everyday use, the term most often refers to coil springs, but there are many different spring designs. Modern springs are typically manufactured from spring steel. An example of a non-metallic spring is the bow, made traditionally of flexible yew wood, which when drawn stores energy to propel an arrow.

An elastic modulus is the unit of measurement of an object's or substance's resistance to being deformed elastically when a stress is applied to it.

<span class="mw-page-title-main">Torsion spring</span> Type of spring

A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional to the amount (angle) it is twisted. There are various types:

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

<span class="mw-page-title-main">Buckling</span> Sudden change in shape of a structural component under load

In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.

<span class="mw-page-title-main">Simple shear</span> Translation which preserves parallelism

Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other.

<span class="mw-page-title-main">Rheometer</span> Scientific instrument used to measure fluid flow (rheology)

A rheometer is a laboratory device used to measure the way in which a viscous fluid flows in response to applied forces. It is used for those fluids which cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer. It measures the rheology of the fluid.

The second polar moment of area, also known as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (deflection), in objects with an invariant cross-section and no significant warping or out-of-plane deformation. It is a constituent of the second moment of area, linked through the perpendicular axis theorem. Where the planar second moment of area describes an object's resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object's resistance to deflection when subjected to a moment applied in a plane perpendicular to the object's central axis. Similar to planar second moment of area calculations, the polar second moment of area is often denoted as . While several engineering textbooks and academic publications also denote it as or , this designation should be given careful attention so that it does not become confused with the torsion constant, , used for non-cylindrical objects.

<span class="mw-page-title-main">Torsion constant</span> Geometrical property of a bars cross-section

The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness. The SI unit for torsion constant is m4.

<span class="mw-page-title-main">Flexural modulus</span> Intensive property in mechanics

In mechanics, the flexural modulus or bending modulus is an intensive property that is computed as the ratio of stress to strain in flexural deformation, or the tendency for a material to resist bending. It is determined from the slope of a stress-strain curve produced by a flexural test, and uses units of force per area. The flexural modulus defined using the 2-point (cantilever) and 3-point bend tests assumes a linear stress strain response.

<span class="mw-page-title-main">Deflection (engineering)</span> Degree to which part of a structural element is displaced under a given load

In structural engineering, deflection is the degree to which a part of a long structural element is deformed laterally under a load. It may be quantified in terms of an angle or a distance . A longitudinal deformation is called elongation.

<span class="mw-page-title-main">Structural engineering theory</span>

Structural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully structural engineers will need a detailed knowledge of mathematics and of relevant empirical and theoretical design codes. They will also need to know about the corrosion resistance of the materials and structures, especially when those structures are exposed to the external environment.

<span class="mw-page-title-main">Atomic force acoustic microscopy</span>

Atomic force acoustic microscopy (AFAM) is a type of scanning probe microscopy (SPM). It is a combination of acoustics and atomic force microscopy. The principal difference between AFAM and other forms of SPM is the addition of a transducer at the bottom of the sample which induces longitudinal out-of-plane vibrations in the specimen. These vibrations are sensed by a cantilever and tip called a probe. The figure shown here is the clear schematic of AFAM principle here B is the magnified version of the tip and sample placed on the transducer and tip having some optical coating generally gold coating to reflect the laser light on to the photodiode.

References

  1. 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"
  2. Martin Wenham (2001), "Stiffness and flexibility", 200 science investigations for young students, SAGE Publications, p. 126, ISBN   978-0-7619-6349-3
  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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.