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Continuum mechanics  

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In physics, elasticity (from Greek ἐλαστός "ductible") is the tendency of solid materials to return to their original shape after being deformed. 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/in^{2}). 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.
When an elastic material is deformed due to an external force, it experiences internal forces that oppose the deformation and restore 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 stiffness 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 uniform extension, whereas the shear modulus applies to shearing.
The elasticity of materials is described by a stressstrain curve, which shows the relation between stress (the average restorative internal force per unit area) and strain (the relative deformation).^{[1]} For most metals or crystalline materials, the curve is linear for small deformations, and so the stressstrain relationship can adequately be described by Hooke's law, and higherorder terms can be ignored. However, for larger stresses beyond the elastic limit, the relation is no longer linear. 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.^{[2]} For rubberlike materials such as elastomers, the gradient of the stressstrain 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.^{[3]} Elasticity is not exhibited only by solids; nonNewtonian fluids, such as viscoelastic fluids, will also exhibit elasticity in certain conditions. 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 stressstrain 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.
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 stress and strain measures are related by a linear relation known as Hooke's law. Linear elasticity describes the behavior of such materials. Cauchy elastic materials and Hypoelastic materials are models that extend Hooke's law to allow for the possibility of large rotations.
For more general situations, any of a number of stress measures can be used provided they are work conjugate to an appropriate finite strain measure, i.e., the product of the stress measure and the strain measure should be equal to the internal energy (which does not depend on how the stress or strain are measured). Hyperelasticity is the preferred approach for dealing with finite strains and several material models analogous to Hooke's law are in use.
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 geometrydependent version of the idea^{[4]} 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",^{[5]}^{[6]}^{[7]} a linear relationship commonly referred to as Hooke's law. This law can be stated as a relationship between force F and displacement x,
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 :
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, systems that exhibit symmetry, such as a onedimensional rod, can often be reduced to applications of Hooke's law.
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.
A material is said to be Cauchyelastic if the Cauchy stress tensor σ is a function of the strain tensor (deformation gradient) F alone:
Even though the stress in a Cauchyelastic material depends only on the state of deformation, the work done by stresses may depend on the path of deformation. Therefore a Cauchy elastic material has a nonconservative structure, and the stress cannot be derived from a scalar "elastic potential" function.
Hypoelastic materials are described by a relation of the form
This model is an extension of linear elasticity and suffers from the same form of nonconservative behaviour as Cauchy elastic materials.
Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a strain energy density function (W). The stressstrain relation for such materials takes the form
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 elastomerbased objects such as gaskets and of biological materials such as soft tissues and cell membranes.
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,^{[8]} indicating that the presence of cracks makes bodies brittler. Microscopically, the stressstrain 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 energyelastic and entropyelastic. 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.^{[9]} 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.^{[10]}
