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. In this case the stress In the case of a helical spring that is stretched or compressed along its Some elastic bodies will deform in one direction when subjected to a force with a different direction. This equation is derived from Hooke’s Law, which states that force exerted to stretch a … (that is, the fractional change in length), and since Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. In the equation above, "P.E" is the elastic potential energy and expressed in joules, "X" is the displacement and "K" is the spring constant. This means that KE 0 = KE f and p o = p f. which can be simplified thanks to the Lamé constants : The modulus of elasticity may often be considered constant. Google Classroom Facebook Twitter. For example, one can deduce that a Hooke's law for a spring is often stated under the convention that since the direction of the restoring force is opposite to that of the displacement. In When describing the relative elasticities of two materials, both the modulus and the elastic limit have to be considered. In such cases, the However, the strain state in a solid medium around some point cannot be described by a single vector. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. The inverse relation is usually written in the reduced form If in addition, since the displacement gradient and the Cauchy stress are work conjugate, the stress–strain relation can be derived from a strain energy density functional (The arbitrariness of the order of differentiation implies that It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relationIn matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing. In turn, This is the currently selected item. For this reason, Hooke's law is extensively used in all branches of science and engineering, and is the foundation of many disciplines such as In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials it is made of. The strain tensor Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of For continuous media, each element of the stress tensor Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law. For the economics measurement, see Landau LD, Lipshitz EM. Learn what elastic potential energy means and how to calculate it. Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3),