Spring Constant
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Hooke's law is an
empirical law Scientific laws or laws of science are statements, based on reproducibility, repeated experiments or observations, that describe or prediction, predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, a ...
which states that the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
() needed to extend or compress a
spring Spring(s) may refer to: Common uses * Spring (season) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a b ...
), and is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist
Robert Hooke Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
. He first stated the law in 1676 as a Latin
anagram An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, typically using all the original letters exactly once. For example, the word ''anagram'' itself can be rearranged into ''nag a ram'', also the word ...
. He published the solution of his anagram in 1678 as: ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. Hooke's equation holds (to some extent) in many other situations where an
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail 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 limit In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
s are reached. On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the forces and deformations are small enough. 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
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
,
molecular mechanics Molecular mechanics uses classical mechanics to model molecular systems. The Born–Oppenheimer approximation is assumed valid and the potential energy of all systems is calculated as a function of the nuclear coordinates using force fields. Mo ...
and
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
. It is also the fundamental principle behind the
spring scale A spring scale, spring balance or newton meter is a type of mechanical force gauge or weighing scale. It consists of a spring fixed at one end with a hook to attach an object at the other. It works in accordance with Hooke's Law, which states th ...
, the
manometer Pressure measurement is the measurement of an applied force by a fluid (liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pressur ...
, the
galvanometer A galvanometer is an electromechanical measuring instrument for electric current. Early galvanometers were uncalibrated, but improved versions, called ammeters, were calibrated and could measure the flow of current more precisely. A galvanom ...
, and the
balance wheel A balance wheel, or balance, is the timekeeping device used in mechanical watches and small clocks, analogous to the pendulum in a pendulum clock. It is a weighted wheel that rotates back and forth, being returned toward its center position by a ...
of the
mechanical clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and the ...
. The modern
theory of elasticity Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
generalizes Hooke's law to say that the
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
(deformation) of an elastic object or material is proportional to the
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
(a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
) that can be represented by a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
rod with uniform
cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...
will behave like a simple spring when stretched, with a stiffness directly proportional to its cross-section area and inversely proportional to its length.


Formal definition


For linear springs

Consider a simple
helical Helical may refer to: * Helix, the mathematical concept for the shape * Helical engine, a proposed spacecraft propulsion drive * Helical spring, a coilspring * Helical plc, a British property company, once a maker of steel bar stock * Helicoil A t ...
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 (when it is not being stretched). Hooke's law states that F_s = kx or, equivalently, x = \frac where is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with and both negative in that case. According to this formula, the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of the applied force as a function of the displacement will be a straight line passing through the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
, whose
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
is . Hooke's law for a spring is sometimes, but rarely, stated under the convention that is the
restoring force In physics, the restoring force is a force that acts to bring a body to its equilibrium position. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the s ...
exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F_s = -kx since the direction of the restoring force is opposite to that of the displacement.


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
shear Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production Science and technology Engineering *Shear strength (soil), the shear strength ...
ing, rather than stretching or compression, the shearing force and the sideways displacement of the plates obey Hooke's law (for small enough deformations). Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), 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. The law also applies when a stretched steel wire is twisted by pulling on a lever attached to one end. In this case the stress can be taken as the force applied to the lever, and as the distance traveled by it along its circular path. Or, equivalently, one can let be the
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
applied by the lever to the end of the wire, and be the angle by which that end turns. In either case is proportional to (although the constant is different in each case.)


Vector formulation

In the case of a helical spring that is stretched or compressed along its
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if and are defined as
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s, Hooke's
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
still holds and says that the force vector is the elongation vector multiplied by a fixed
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
.


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 (and its value is not too large); 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 In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. More precisely, if e_1,\dots,e_n are elements of a (left) module over a ring ( ...
between the force and deformation vectors, as long as they are small enough. Namely, there is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
from vectors to vectors, such that , and for any real numbers , and any displacement vectors , . Such a function is called a (second-order)
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
. With respect to an arbitrary
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, the force and displacement vectors can be represented by 3 × 1
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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: \mathbf \,=\, \begin F_1\\ F_2 \\ F_3 \end \,=\, \begin \kappa_& \kappa_& \kappa_\\ \kappa_& \kappa_& \kappa_\\ \kappa_& \kappa_& \kappa_ \end \begin X_1\\ X_2 \\ X_3 \end \,=\, \boldsymbol \mathbf That is, F_i = \kappa_ X_1 + \kappa_ X_2 + \kappa_ X_3 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
elastic material (such as a block of rubber, the wall of a
boiler A boiler is a closed vessel in which fluid (generally water) is heated. The fluid does not necessarily boil. The heated or vaporized fluid exits the boiler for use in various processes or heating applications, including water heating, central h ...
, or a steel bar) 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 In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
(in lieu of the displacement ) and the stress tensor (replacing the restoring force ). The analogue of Hooke's spring law for continuous media is then \boldsymbol = \mathbf \boldsymbol, where is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the
stiffness tensor Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a b ...
or
elasticity tensor 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 ...
. One may also write it as \boldsymbol = \mathbf \boldsymbol, 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 \boldsymbol \,=\, \begin \varepsilon_ & \varepsilon_ & \varepsilon_\\ \varepsilon_ & \varepsilon_ & \varepsilon_\\ \varepsilon_ & \varepsilon_ & \varepsilon_ \end \,;\qquad \boldsymbol \,=\, \begin \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \end 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 \sigma_ = \sum_^3 \sum_^3 c_ \varepsilon_ 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 Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, and
microstructure Microstructure is the very small scale structure of a material, defined as the structure of a prepared surface of material as revealed by an optical microscope above 25× magnification. The microstructure of a material (such as metals, polymers ...
. 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 In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a r ...
crystal, 5 for an
hexagonal In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
structure, and 3 for a cubic symmetry. For
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
media (which have the same physical properties in any direction), can be reduced to only two independent numbers, the
bulk modulus The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other moduli describe ...
and the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel ...
, 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
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s, or the polarization of a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
by an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
. In particular, the tensor equation relating elastic stresses to strains is entirely similar to the equation relating the
viscous stress tensor The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress ...
and the
strain rate tensor In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defi ...
in flows of
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
fluids; although the former pertains to
static Static may refer to: Places *Static Nunatak, a nunatak in Antarctica United States * Static, Kentucky and Tennessee *Static Peak, a mountain in Wyoming **Static Peak Divide, a mountain pass near the peak Science and technology Physics *Static el ...
stresses (related to ''amount'' of deformation) while the latter pertains to
dynamical In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
stresses (related to the ''rate'' of deformation).


Units of measurement

In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, displacements are measured in meters (m), and forces in
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
s (N or kg·m/s2). Therefore, the spring constant , and each element of the tensor , is measured in newtons per meter (N/m), or kilograms per second squared (kg/s2). 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
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Fren ...
s (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor are
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
(displacements divided by distances). Therefore, the entries of are also expressed in units of pressure.


General application to elastic materials

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. Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic range (i.e., for stresses below the
yield strength In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials a
proportional limit In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
stress is defined, below which the errors associated with the linear approximation are negligible. Rubber is generally regarded as a "non-Hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate. Generalizations of Hooke's law for the case of large deformations is provided by models of
neo-Hookean solid A neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In c ...
s and
Mooney–Rivlin solid In continuum mechanics, a Mooney–Rivlin solidMooney, M., 1940, ''A theory of large elastic deformation'', Journal of Applied Physics, 11(9), pp. 582–592.Rivlin, R. S., 1948, ''Large elastic deformations of isotropic materials. IV. Further deve ...
s.


Derived formulae


Tensional stress of a uniform bar

A rod of any
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
material may be viewed as a linear
spring Spring(s) may refer to: Common uses * Spring (season), a season of the year * Spring (device), a mechanical device that stores energy * Spring (hydrology), a natural source of water * Spring (mathematics), a geometric surface in the shape of a ...
. The rod has length and cross-sectional area . Its
tensile stress In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
is linearly proportional to its fractional extension or strain by the
modulus of elasticity An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
: \sigma = E \varepsilon. The modulus of elasticity may often be considered constant. In turn, \varepsilon = \frac (that is, the fractional change in length), and since \sigma = \frac \,, it follows that: \varepsilon = \frac = \frac\,. The change in length may be expressed as \Delta L = \varepsilon L = \frac\,.


Spring energy

The potential energy stored in a spring is given by U_\mathrm(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative. This potential can be visualized as a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
on the -plane such that . As the spring is stretched in the positive -direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate: \frac=k\,. Note that the change in the change in is constant even when the displacement and acceleration are zero.


Relaxed force constants (generalized compliance constants)

Relaxed force constants (the inverse of generalized
compliance constants Compliance Constants are the elements of an inverted Hessian matrix. The calculation of compliance constants provides an alternative description of chemical bonds in comparison with the widely used force constants explicitly ruling out the depende ...
) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for
reactant In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
s,
transition state In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked wi ...
s, and products of a
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
. Just as the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed compliance constants. A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis. The suitability of relaxed force constants (inverse compliance constants) as
covalent bond A covalent bond is a chemical bond that involves the sharing of electrons to form electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs. The stable balance of attractive and repulsive forces between atoms ...
strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.


Harmonic oscillator

A mass attached to the end of a spring is a classic example of a
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
. By pulling slightly on the mass and then releasing it, the system will be set in
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
and the mass of the spring, the amplitude of the oscillation will remain constant; and its
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
will be independent of its amplitude, determined only by the mass and the stiffness of the spring: f = \frac \sqrt\frac This phenomenon made possible the construction of accurate
mechanical clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and the ...
s and watches that could be carried on ships and people's pockets.


Rotation in gravity-free space

If the mass were attached to a spring with force constant and rotating in free space, the spring tension () would supply the required
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
(): F_\mathrm = kx\,; \qquad F_\mathrm = m \omega^2 r Since and , then: k = m \omega^2 Given that , this leads to the same frequency equation as above: f = \frac \sqrt\frac


Linear elasticity theory for continuous media


Isotropic materials

Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor. Thus in
index notation In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to th ...
: \varepsilon_ = \left(\tfrac13\varepsilon_\delta_\right) + \left(\varepsilon_-\tfrac13\varepsilon_\delta_\right) where is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. In direct tensor notation: \boldsymbol = \operatorname(\boldsymbol) + \operatorname(\boldsymbol) \,; \qquad \operatorname(\boldsymbol) = \tfrac13\operatorname(\boldsymbol)~\mathbf \,; \qquad \operatorname(\boldsymbol) = \boldsymbol - \operatorname(\boldsymbol) where is the second-order identity tensor. The first term on the right is the constant tensor, also known as the volumetric strain tensor, and the second term is the traceless symmetric tensor, also known as the deviatoric strain tensor or
shear tensor Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production Science and technology Engineering *Shear strength (soil), the shear strength ...
. The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors: \sigma_=3K\left(\tfrac\varepsilon_\delta_\right) +2G\left(\varepsilon_-\tfrac\varepsilon_\delta_\right)\,; \qquad \boldsymbol = 3K\operatorname(\boldsymbol) + 2G\operatorname(\boldsymbol) where is the
bulk modulus The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other moduli describe ...
and is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel ...
. Using the relationships between the
elastic moduli An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is \boldsymbol = \lambda\operatorname(\boldsymbol)\mathbf + 2\mu\boldsymbol = \mathsf:\boldsymbol \,; \qquad \mathsf = \lambda\mathbf\otimes\mathbf + 2\mu\mathsf where and are the
Lamé constants Lamé may refer to: *Lamé (fabric), a clothing fabric with metallic strands *Lamé (fencing), a jacket used for detecting hits * Lamé (crater) on the Moon * Ngeté-Herdé language, also known as Lamé, spoken in Chad *Peve language, also known ...
, is the second-rank identity tensor, and I is the symmetric part of the fourth-rank identity tensor. In index notation: \sigma_ = \lambda\varepsilon_~\delta_ + 2\mu\varepsilon_ = c_\varepsilon_ \,;\qquad c_ = \lambda\delta_\delta_ + \mu\left(\delta_\delta_ + \delta_\delta_\right) The inverse relationship is \boldsymbol = \frac\boldsymbol - \frac\operatorname(\boldsymbol)\mathbf = \frac \boldsymbol + \left(\frac - \frac\right)\operatorname(\boldsymbol)\mathbf Therefore, the compliance tensor in the relation is \mathsf = - \frac\mathbf\otimes\mathbf + \frac\mathsf = \left(\frac - \frac\right)\mathbf\otimes\mathbf + \frac\mathsf In terms of
Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
and
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
, Hooke's law for isotropic materials can then be expressed as \varepsilon_=\frac\big(\sigma_-\nu(\sigma_\delta_-\sigma_)\big) \,; \qquad \boldsymbol = \frac \big(\boldsymbol - \nu(\operatorname(\boldsymbol)\mathbf - \boldsymbol)\big) = \frac\boldsymbol - \frac\operatorname(\boldsymbol)\mathbf This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is \begin \varepsilon_ & = \frac \big(\sigma_ - \nu(\sigma_+\sigma_) \big) \\ \varepsilon_ & = \frac \big(\sigma_ - \nu(\sigma_+\sigma_) \big) \\ \varepsilon_ & = \frac \big(\sigma_ - \nu(\sigma_+\sigma_) \big) \\ \varepsilon_ & = \frac \sigma_ \,;\qquad \varepsilon_ = \frac\sigma_ \,;\qquad \varepsilon_ = \frac\sigma_ \end where is
Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
and is
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
. (See 3-D elasticity). In matrix form, Hooke's law for isotropic materials can be written as \begin\varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end \,=\, \begin\varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ \gamma_ \\ \gamma_ \\ \gamma_ \end \,=\, \frac \begin 1 & -\nu & -\nu & 0 & 0 & 0 \\ -\nu & 1 & -\nu & 0 & 0 & 0 \\ -\nu & -\nu & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2+2\nu & 0 & 0 \\ 0 & 0 & 0 & 0 & 2+2\nu & 0 \\ 0 & 0 & 0 & 0 & 0 & 2+2\nu \end \begin\sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end where is the engineering shear strain. The inverse relation may be written as \begin\sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end \,=\, \frac \begin 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac \end \begin\varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end which can be simplified thanks to the Lamé constants: \begin\sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end \,=\, \begin 2\mu+\lambda & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & 2\mu+\lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & 2\mu+\lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end \begin \varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end In vector notation this becomes \begin \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \end \,=\, 2\mu \begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \end + \lambda \mathbf\left(\varepsilon_ + \varepsilon_ + \varepsilon_ \right) where is the identity tensor.


Plane stress

Under
plane stress In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analys ...
conditions, . In that case Hooke's law takes the form \begin\sigma_ \\ \sigma_ \\ \sigma_ \end \,=\, \frac \begin 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac \end \begin\varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \end In vector notation this becomes \begin \sigma_ & \sigma_ \\ \sigma_ & \sigma_ \end \,=\, \frac \left((1-\nu) \begin \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ \end + \nu \mathbf \left(\varepsilon_ + \varepsilon_ \right) \right) The inverse relation is usually written in the reduced form \begin\varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \end \,=\, \frac \begin 1 & -\nu & 0 \\ -\nu & 1 & 0 \\ 0 & 0 & 2+2\nu \end \begin\sigma_ \\ \sigma_ \\ \sigma_ \end


Plane strain

Under
plane strain In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
conditions, . In this case Hooke's law takes the form \begin\sigma_ \\ \sigma_ \\ \sigma_ \end \,=\, \frac \begin 1 - \nu & \nu & 0 \\ \nu & 1 - \nu & 0 \\ 0 & 0 & \frac \end \begin\varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \end


Anisotropic materials

The symmetry of the
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
() and the generalized Hooke's laws () implies that . Similarly, the symmetry of the infinitesimal strain tensor implies that . These symmetries are called the minor symmetries of the stiffness tensor c. This reduces the number of elastic constants from 81 to 36. 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 (), then \sigma_ = \frac \quad \implies \quad c_ = \frac\,. The arbitrariness of the order of differentiation implies that . These are called the major symmetries of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components.


Matrix representation (stiffness tensor)

It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called
Voigt notation In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notat ...
. To do this we take advantage of the symmetry of the stress and strain tensors and express them as six-dimensional vectors in an orthonormal coordinate system () as boldsymbol\,=\, \begin\sigma_\\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end \,\equiv\, \begin \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end \,;\qquad boldsymbol\,=\, \begin\varepsilon_\\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end \,\equiv\, \begin \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end Then the stiffness tensor (c) can be expressed as mathsf\,=\, \begin c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \end \,\equiv\, \begin C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \end and Hooke's law is written as boldsymbol= mathsfboldsymbol\qquad \text \qquad \sigma_i = C_ \varepsilon_j \,. Similarly the compliance tensor (s) can be written as mathsf\,=\, \begin s_ & s_ & s_ & 2s_ & 2s_ & 2s_ \\ s_ & s_ & s_ & 2s_ & 2s_ & 2s_ \\ s_ & s_ & s_ & 2s_ & 2s_ & 2s_ \\ 2s_ & 2s_ & 2s_ & 4s_ & 4s_ & 4s_ \\ 2s_ & 2s_ & 2s_ & 4s_ & 4s_ & 4s_ \\ 2s_ & 2s_ & 2s_ & 4s_ & 4s_ & 4s_ \end \,\equiv\, \begin S_ & S_ & S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ & S_ & S_ \end


Change of coordinate system

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 relation c_ = l_l_l_l_c_ where are the components of an orthogonal rotation matrix . The same relation also holds for inversions. In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by mathbf_i'= \mathbf_i] then C_\varepsilon_i\varepsilon_j = C_'\varepsilon'_i\varepsilon'_j \,. In addition, if the material is symmetric with respect to the transformation then C_ = C'_ \quad \implies \quad C_(\varepsilon_i\varepsilon_j - \varepsilon'_i\varepsilon'_j) = 0 \,.


Orthotropic materials

Orthotropic material In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength can b ...
s have three
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
planes of symmetry. If the basis vectors () are normals to the planes of symmetry then the coordinate transformation relations imply that \begin \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end \,=\, \begin C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & 0 & C_ \end \begin \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end The inverse of this relation is commonly written as \begin \varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end \,=\, \begin \frac & - \frac & - \frac & 0 & 0 & 0 \\ -\frac & \frac & - \frac & 0 & 0 & 0 \\ -\frac & - \frac & \frac & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac \\ \end \begin \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end where * is the
Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
along axis * is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel ...
in direction on the plane whose normal is in direction * is the
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
that corresponds to a contraction in direction when an extension is applied in direction . Under ''plane stress'' conditions, , Hooke's law for an orthotropic material takes the form \begin\varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \end \,=\, \begin \frac & -\frac & 0 \\ -\frac & \frac & 0 \\ 0 & 0 & \frac \end \begin\sigma_ \\ \sigma_ \\ \sigma_ \end \,. The inverse relation is \begin\sigma_ \\ \sigma_ \\ \sigma_ \end \,=\, \frac \begin E_ & \nu_E_ & 0 \\ \nu_E_ & E_ & 0 \\ 0 & 0 & G_(1-\nu_\nu_) \end \begin\varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \end \,. The transposed form of the above stiffness matrix is also often used.


Transversely isotropic materials

A
transversely isotropic A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties ar ...
material is symmetric with respect to a rotation about an
axis of symmetry Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.
. For such a material, if is the axis of symmetry, Hooke's law can be expressed as \begin \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end \,=\, \begin C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac \end \begin \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end More frequently, the axis is taken to be the axis of symmetry and the inverse Hooke's law is written as \begin \varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end \,=\, \begin \frac & - \frac & - \frac & 0 & 0 & 0 \\ -\frac & \frac & - \frac & 0 & 0 & 0 \\ -\frac & - \frac & \frac & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac \\ \end \begin \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end :


Universal elastic anisotropy index

To grasp the degree of anisotropy of any class, a universal elastic anisotropy index (AU) was formulated. It replaces the
Zener ratio The Zener ratio is a dimensionless number that is used to quantify the anisotropy for Cubic crystal system, cubic crystals. It is sometimes referred as ''anisotropy ratio'' and is named after Clarence Zener. Conceptually, it quantifies how far a mat ...
, which is suited for cubic crystals.


Thermodynamic basis

Linear deformations of elastic materials can be approximated as adiabatic. Under these conditions and for quasistatic processes the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amoun ...
for a deformed body can be expressed as \delta W = \delta U where is the increase in
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
and is the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal tr ...
done by external forces. The work can be split into two terms \delta W = \delta W_\mathrm + \delta W_\mathrm where is the work done by
surface force Surface force denoted ''fs'' is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed into two perpendicular components: normal forces and shear forces. A normal force acts normal ...
s while is the work done by
body force In physics, a body force is a force that acts throughout the volume of a body. Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s. If is a variation of the displacement field in the body, then the two external work terms can be expressed as \delta W_\mathrm = \int_ \mathbf\cdot\delta\mathbf\,dS \,; \qquad \delta W_\mathrm = \int_ \mathbf\cdot\delta\mathbf\,dV where is the surface traction vector, is the body force vector, represents the body and represents its surface. Using the relation between the Cauchy stress and the surface traction, (where is the unit outward normal to ), we have \delta W = \delta U = \int_ (\mathbf\cdot\boldsymbol)\cdot\delta\mathbf\,dS + \int_ \mathbf\cdot\delta\mathbf\,dV\,. Converting the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
into a
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ap ...
via the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
gives \delta U = \int_ \big(\nabla\cdot(\boldsymbol\cdot\delta\mathbf) + \mathbf\cdot\delta\mathbf\big)\, dV \,. Using the symmetry of the Cauchy stress and the identity \nabla\cdot(\mathbf\cdot\mathbf) = (\nabla\cdot\mathbf)\cdot\mathbf+\tfrac12\left(\mathbf^\mathsf : \nabla\mathbf+ \mathbf:(\nabla\mathbf)^\mathsf\right) we have the following \delta U = \int_ \left(\boldsymbol:\tfrac12\left(\nabla\delta\mathbf+(\nabla\delta\mathbf)^\mathsf\right) + \left(\nabla\cdot\boldsymbol+\mathbf\right)\cdot\delta\mathbf\right)\,dV \,. From the definition of
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
and from the equations of equilibrium we have \delta\boldsymbol = \tfrac12\left(\nabla\delta\mathbf+(\nabla\delta\mathbf)^\mathsf\right) \,;\qquad \nabla\cdot\boldsymbol+\mathbf=\mathbf \,. Hence we can write \delta U = \int_ \boldsymbol:\delta\boldsymbol\,dV and therefore the variation in the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
density is given by \delta U_0 = \boldsymbol:\delta\boldsymbol \,. An
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
material is defined as one in which the total internal energy is equal to the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of the internal forces (also called the elastic strain energy). Therefore, the internal energy density is a function of the strains, and the variation of the internal energy can be expressed as \delta U_0 = \frac:\delta\boldsymbol \,. Since the variation of strain is arbitrary, the stress–strain relation of an elastic material is given by \boldsymbol = \frac\,. For a linear elastic material, the quantity is a linear function of , and can therefore be expressed as \boldsymbol = \mathsf:\boldsymbol where c is a fourth-rank tensor of material constants, also called the stiffness tensor. We can see why c must be a fourth-rank tensor by noting that, for a linear elastic material, \frac\boldsymbol(\boldsymbol) = \text = \mathsf \,. In index notation \frac = \text = c_ \,. The right-hand side constant requires four indices and is a fourth-rank quantity. 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. We can also show that the constant obeys the tensor transformation rules for fourth-rank tensors.


See also

*
Acoustoelastic effect The acoustoelastic effect is how the sound velocities (both longitudinal and shear wave velocities) of an elastic material change if subjected to an initial static stress field. This is a non-linear effect of the constitutive relation between mec ...
*
Elastic potential energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, s ...
*
Laws of science Scientific laws or laws of science are statements, based on reproducibility, repeated experiments or observations, that describe or prediction, predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, a ...
* List of scientific laws named after people * Quadratic form * Series and parallel springs * Spring system * Simple harmonic motion#Mass on a spring, Simple harmonic motion of a mass on a spring * Sine wave * Solid mechanics * Spring pendulum


Notes


References

* * :File:hookeslaw.ogg, Walter Lewin explains Hooke's law. From * :File:hookeslawexample.ogg, A test of Hooke's law. From


External links


JavaScript Applet demonstrating Springs and Hooke's law

JavaScript Applet demonstrating Spring Force
{{Elastic moduli 1660 in science Springs (mechanical) Elasticity (physics) Solid mechanics Structural analysis