Elasticity (physics)

In physics, elasticity (from Greek ἐλαστός "ductible") is the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. Solid objects will deform when forces are applied on them. If the material is elastic, the object will return to its initial shape and size when these forces are removed.

The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.

Perfect elasticity is an approximation of the real world, and few materials remain purely elastic even after very small deformations. In engineering, the amount of elasticity of a material is determined by two types of material parameter. The first type of material parameter is called a modulus, which measures the amount of force per unit area (stress) needed to achieve a given amount of deformation. The units of modulus are pascals (Pa) or pounds of force per square inch (psi, also lbf/in²). A higher modulus typically indicates that the material is harder to deform. The second type of parameter measures the elastic limit. The limit can be a stress beyond which the material no longer behaves elastic and deformation of the material will take place. If the stress is released, the material will elastically return to a permanent deformed shape instead of the original shape.

When describing the relative elasticities of two materials, both the modulus and the elastic limit have to be considered. Rubbers typically have a low modulus and tend to stretch a lot (that is, they have a high elastic limit) and so appear more elastic than metals (high modulus and low elastic limit) in everyday experience. Of two rubber materials with the same elastic limit, the one with a lower modulus will appear to be more elastic.

Overview

When an elastic material is deformed due to an external force, it experiences internal resistance to the deformation and restores it to its original state if the external force is no longer applied. There are various elastic moduli, such as Young's modulus, the shear modulus, and the bulk modulus, all of which are measures of the inherent elastic properties of a material as a resistance to deformation under an applied load. The various moduli apply to different kinds of deformation. For instance, Young's modulus applies to extension/compression of a body, whereas the shear modulus applies to its shear.^[1]

The elasticity of materials is described by a stress-strain curve, which shows the relation between stress (the average restorative internal force per unit area) and strain (the relative deformation).^[2] The curve is generally nonlinear, but it can (by use of a Taylor series) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible). If the material is isotropic, the linearized stress-strain relationship is called Hooke's law, which is often presumed to apply up to the elastic limit for most metals or crystalline materials whereas nonlinear elasticity is generally required to model large deformations of rubbery materials even in the elastic range. For even higher stresses, materials exhibit plastic behavior, that is, they deform irreversibly and do not return to their original shape after stress is no longer applied.^[3] For rubber-like materials such as elastomers, the slope of the stress-strain curve increases with stress, meaning that rubbers progressively become more difficult to stretch, while for most metals, the gradient decreases at very high stresses, meaning that they progressively become easier to stretch.^[4] Elasticity is not exhibited only by solids; non-Newtonian fluids, such as viscoelastic fluids, will also exhibit elasticity in certain conditions quantified by the Deborah number. In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, these fluids may start to flow like a viscous liquid.

Because the elasticity of a material is described in terms of a stress-strain relation, it is essential that the terms stress and strain be defined without ambiguity. Typically, two types of relation are considered. The first type deals with materials that are elastic only for small strains. The second deals with materials that are not limited to small strains. Clearly, the second type of relation is more general in the sense that it must include the first type as a special case.

For small strains, the measure of stress that is used is the Cauchy stress while the measure of strain that is used is the infinitesimal strain tensor; the resulting (predicted) material behavior is termed linear elasticity, which (for isotropic media) is called the generalized Hooke's law. Cauchy elastic materials and hypoelastic materials are models that extend Hooke's law to allow for the possibility of large rotations, large distortions, and intrinsic or induced anisotropy.

For more general situations, any of a number of stress measures can be used, and it generally desired (but not required) that the elastic stress-strain relation be phrased in terms of a finite strain measure that is work conjugate to the selected stress measure, i.e., the time integral of the inner product of the stress measure with the rate of the strain measure should be equal to the change in internal energy for any adiabatic process that remains below the elastic limit.

Linear elasticity

As noted above, for small deformations, most elastic materials such as springs exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known as Hooke's law. A geometry-dependent version of the idea^[5] was first formulated by Robert Hooke in 1675 as a Latin anagram, "ceiiinosssttuv". He published the answer in 1678: "Ut tensio, sic vis" meaning "As the extension, so the force",^[6]^[7]^[8] a linear relationship commonly referred to as Hooke's law. This law can be stated as a relationship between tensile force $F$ and corresponding extension displacement $x$ ,

F=kx,

where $k$ is a constant known as the rate or spring constant. It can also be stated as a relationship between stress $σ$ and strain $\varepsilon$ :

\sigma =E\varepsilon ,

where $E$ Is known as the elastic modulus or Young's modulus.

Although the general proportionality constant between stress and strain in three dimensions is a 4th-order tensor called stiffness, systems that exhibit symmetry, such as a one-dimensional rod, can often be reduced to applications of Hooke's law.

Finite elasticity

The elastic behavior of objects that undergo finite deformations have been described using a number of models, such as Cauchy elastic material models, Hypoelastic material models, and Hyperelastic material models. The primary measure that is used to quantity finite strains is the deformation gradient (F). More convenient strain measures can be derived from this primary quantity.

Cauchy elastic materials

A material is said to be Cauchy-elastic if the Cauchy stress tensor σ is a function of the strain tensor (deformation gradient) F alone:

\ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})

Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses may depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "hyperelastic" models (for which stress can be derived from a scalar "elastic potential" function).

Hypoelastic materials

Hypoelastic materials are described by a relation of the form

{\dot {\boldsymbol {\sigma }}}={\mathsf {D}}:{\dot {\boldsymbol {F}}}\,.

This model is an extension of linear elasticity and suffers from the same form of non-conservative behaviour as Cauchy elastic materials.

Hyperelastic materials

Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a strain energy density function (W). The stress-strain relation for such materials takes the form

{\boldsymbol {\sigma }}={\cfrac {1}{J}}~{\cfrac {\partial W}{\partial {\boldsymbol {F}}}}\cdot {\boldsymbol {F}}^{T}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.

Applications

Linear elasticity is used widely in the design and analysis of structures such as beams, plates and shells, and sandwich composites. This theory is also the basis of much of fracture mechanics.

Hyperelasticity is primarily used to determine the response of elastomer-based objects such as gaskets and of biological materials such as soft tissues and cell membranes.

Factors affecting elasticity

For isotropic materials, the presence of fractures affects the Young and the shear modulus perpendicular to the planes of the cracks, which decrease (Young's modulus faster than the shear modulus) as the fracture density increases,^[9] indicating that the presence of cracks makes bodies brittler. Microscopically, the stress-strain relationship of materials is in general governed by the Helmholtz free energy, a thermodynamic quantity. Molecules settle in the configuration which minimizes the free energy, subject to constraints derived from their structure, and, depending on whether the energy or the entropy term dominates the free energy, materials can broadly be classified as energy-elastic and entropy-elastic. As such, microscopic factors affecting the free energy, such as the equilibrium distance between molecules, can affect the elasticity of materials: for instance, in inorganic materials, as the equilibrium distance between molecules at 0 K increases, the bulk modulus decreases.^[10] The effect of temperature on elasticity is difficult to isolate, because there are numerous factors affecting it. For instance, the bulk modulus of a material is dependent on the form of its lattice, its behavior under expansion, as well as the vibrations of the molecules, all of which are dependent on temperature.^[11]

References

^ Landau LD, Lipshitz EM. Theory of Elasticity, 3rd Edition, 1970: 1-172.
^ Treloar, L. R. G. (1975). The Physics of Rubber Elasticity. Oxford: Clarendon Press. p. 2. ISBN 978-0-1985-1355-1.
^ Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 70. ISBN 978-0-1237-4446-3.
^ de With, Gijsbertus (2006). Structure, Deformation, and Integrity of Materials, Volume I: Fundamentals and Elasticity. Weinheim: Wiley VCH. p. 32. ISBN 978-3-527-31426-3.
^ Descriptions of material behavior should be independent of the geometry and shape of the object made of the material under consideration. The original version of Hooke's law involves a stiffness constant that depends on the initial size and shape of the object. The stiffness constant is therefore not strictly a material property.
^ Atanackovic, Teodor M.; Guran, Ardéshir (2000). "Hooke's law". Theory of elasticity for scientists and engineers. Boston, Mass.: Birkhäuser. p. 85. ISBN 978-0-8176-4072-9.
^ "Strength and Design". Centuries of Civil Engineering: A Rare Book Exhibition Celebrating the Heritage of Civil Engineering. Linda Hall Library of Science, Engineering & Technology.^{[page needed]}
^ Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, 2012 . ISBN 9781107025417.^{[page needed]}
^ Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 387. ISBN 978-0-1237-4446-3.
^ Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 344. ISBN 978-0-1237-4446-3.
^ Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 365. ISBN 978-0-1237-4446-3.

[1] Landau LD, Lipshitz EM. Theory of Elasticity, 3rd Edition, 1970: 1-172.

[2] Treloar, L. R. G. (1975). The Physics of Rubber Elasticity. Oxford: Clarendon Press. p. 2. ISBN 978-0-1985-1355-1.

[3] Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 70. ISBN 978-0-1237-4446-3.

[4] With, Gijsbertus (2006). Structure, Deformation, and Integrity of Materials, Volume I: Fundamentals and Elasticity. Weinheim: Wiley VCH. p. 32. ISBN 978-3-527-31426-3.

[5] Descriptions of material behavior should be independent of the geometry and shape of the object made of the material under consideration. The original version of Hooke's law involves a stiffness constant that depends on the initial size and shape of the object. The stiffness constant is therefore not strictly a material property.

[6] Atanackovic, Teodor M.; Guran, Ardéshir (2000). "Hooke's law". Theory of elasticity for scientists and engineers. Boston, Mass.: Birkhäuser. p. 85. ISBN 978-0-8176-4072-9.

[7] "Strength and Design". Centuries of Civil Engineering: A Rare Book Exhibition Celebrating the Heritage of Civil Engineering. Linda Hall Library of Science, Engineering & Technology.^{[page needed]}

[8] Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, 2012 . ISBN 9781107025417.^{[page needed]}

[9] Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 387. ISBN 978-0-1237-4446-3.

[10] Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 344. ISBN 978-0-1237-4446-3.

[11] Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 365. ISBN 978-0-1237-4446-3.

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