Hooke's law
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, where is a constant factor characteristic of the spring, and 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. It is the fundamental principle behind the spring scale, the manometer, the galvanometer, and the balance wheel of the mechanical clock.
The equation holds in many situations where an elastic body is deformed. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hooke's law is a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It fails once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.
Definition
The modern theory of elasticity generalizes Hooke's law to say that the strain of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map that can be represented by a matrix of real numbers.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 they are made of. For example, one can deduce that a homogeneous rod with uniform cross section will behave like a simple spring when stretched, with a stiffness directly proportional to its cross-section area and inversely proportional to its length.
Linear springs
Consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let be the amount by which the free end of the spring was displaced from its "relaxed" position. Hooke's law states that or, equivalently,where is a positive real number, characteristic of the spring. A spring with spaces between the coils can be compressed, and the same formula holds for compression, with and both negative in that case.
According to this formula, the graph of the applied force as a function of the displacement will be a straight line passing through the origin, whose slope is.
Hooke's law for a spring is also stated under the convention that is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes since the direction of the restoring force is opposite to that of the displacement.
Torsional springs
The torsional analog of Hooke's law applies to torsional springs. It states that the torque required to rotate an object is directly proportional to the angular displacement from the equilibrium position. It describes the relationship between the torque applied to an object and the resulting angular deformation due to torsion. Mathematically, it can be expressed as:Where:
- τ is the torque measured in Newton-meters or N·m.
- k is the torsional constant, which characterizes the stiffness of the torsional spring or the resistance to angular displacement.
- θ is the angular displacement from the equilibrium position.
General "scalar" springs
Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative.For example, when a block of rubber attached to two parallel plates is deformed by shearing, rather than stretching or compression, the shearing force and the sideways displacement of the plates obey Hooke's law.
Hooke's law also applies when a straight steel bar or concrete beam, supported at both ends, is bent by a weight placed at some intermediate point. The displacement in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.
Vector formulation
In the case of a helical spring that is stretched or compressed along its axis, the applied force and the resulting elongation or compression have the same direction. Therefore, if and are defined as vectors, Hooke's equation still holds and says that the force vector is the elongation vector multiplied by a fixed scalar.General tensor form
Some elastic bodies will deform in one direction when subjected to a force with a different direction. 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. In such cases, the magnitude of the displacement will be proportional to the magnitude of the force, as long as the direction of the latter remains the same ; so the scalar version of Hooke's law will hold. However, the force and displacement vectors will not be scalar multiples of each other, since they have different directions. Moreover, the ratio between their magnitudes will depend on the direction of the vector.Yet, in such cases there is often a fixed linear relation between the force and deformation vectors, as long as they are small enough. Namely, there is a function from vectors to vectors, such that, and for any real numbers, and any displacement vectors,. Such a function is called a tensor.
With respect to an arbitrary Cartesian coordinate system, the force and displacement vectors can be represented by 3 × 1 matrices of real numbers. Then the tensor connecting them can be represented by a 3 × 3 matrix of real coefficients, that, when multiplied by the displacement vector, gives the force vector:
That is, for. Therefore, Hooke's law can be said to hold also when and are vectors with variable directions, except that the stiffness of the object is a tensor, rather than a single real number.
Hooke's law for continuous media
The stresses and strains of the material inside a continuous elastic material are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.
In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the strain tensor and the stress tensor . The analogue of Hooke's spring law for continuous media is then where is a fourth-order tensor usually called the stiffness tensor or elasticity tensor. One may also write it as where the tensor, called the compliance tensor, represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
Being a linear mapping between the nine numbers and the nine numbers, the stiffness tensor is represented by a matrix of real numbers. Hooke's law then says that
where.
All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, pressure, and microstructure.
Due to the inherent symmetries of,, and, only 21 elastic coefficients of the latter are independent. This number can be further reduced by the symmetry of the material: 9 for an orthorhombic crystal, 5 for an hexagonal structure, and 3 for a cubic symmetry. For isotropic media, can be reduced to only two independent numbers, the bulk modulus and the shear modulus, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.
Analogous laws
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 fluids, or the polarization of a dielectric by an electric field.In particular, the tensor equation relating elastic stresses to strains is entirely similar to the equation relating the viscous stress tensor and the strain rate tensor in flows of viscous fluids; although the former pertains to static stresses while the latter pertains to dynamical stresses.
Units of measurement
In SI units, displacements are measured in meters, and forces in newtons. Therefore, the spring constant, and each element of the tensor, is measured in newtons per meter, or kilograms per second squared.For continuous media, each element of the stress tensor is a force divided by an area; it is therefore measured in units of pressure, namely pascals. The elements of the strain tensor are dimensionless. Therefore, the entries of are also expressed in units of pressure.