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In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
Note that not all elastic materials undergo linear elastic deformation; some, such as concrete, gray cast iron, and many polymers, respond in a nonlinear fashion. For these materials Hooke's law is inapplicable. [2] Difference in true and engineering stress-strain curves
Hooke's law states that the force required to deform elastic objects should be directly proportional to the distance of deformation, regardless of how large that distance becomes. This is known as perfect elasticity , in which a given object will return to its original shape no matter how strongly it is deformed.
In a continuous body, a deformation field results from a stress field due to applied forces or because of some changes in the conditions of the body. The relation between stress and strain (relative deformation) is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.
Anelasticity is a property of materials that describes their behaviour when undergoing deformation.Its formal definition does not include the physical or atomistic mechanisms but still interprets the anelastic behaviour as a manifestation of internal relaxation processes.
The elastic components, as previously mentioned, can be modeled as springs of elastic constant E, given the formula: = where σ is the stress, E is the elastic modulus of the material, and ε is the strain that occurs under the given stress, similar to Hooke's law.
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
The isochoric deformation gradient is defined as ... These consistency conditions can be found by comparing Hooke's law with linearized hyperelasticity at small strains.