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Catalysis

Catalysis is the increase in rate of a chemical reaction due to an added substance known as a catalyst. Catalysts are not consumed by the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recycles quickly, very small amounts of catalyst often suffice; mixing, surface area, and temperature are important factors in reaction rate. Catalysts generally react with one or more reactants to form intermediates that subsequently give the final reaction product, in the process of regenerating the catalyst.

The rate increase occurs because the catalyst allows the reaction to occur by an alternative mechanism which may be much faster than the non-catalyzed mechanism. However the non-catalyzed mechanism does remain possible, so that the total rate (catalyzed plus non-catalyzed) can only increase in the presence of the catalyst and never decrease.

Catalysis may be classified as either homogeneous, whose components are dispersed in the same phase (usually gaseous or liquid) as the reactant, or heterogeneous, whose components are not in the same phase. Enzymes and other biocatalysts are often considered as a third category.

Catalysis is ubiquitous in chemical industry of all kinds. Estimates are that 90% of all commercially produced chemical products involve catalysts at some stage in the process of their manufacture.

The term "catalyst" is derived from Greek καταλύειν, kataluein, meaning "loosen" or "untie". The concept of catalysis was invented by chemist Elizabeth Fulhame, based on her novel work in oxidation-reduction experiments.

Example
An illustrative example is the effect of catalysts to speed the decomposition of hydrogen peroxide into water and oxygen:
 * 2 H$2$O$2$ →   2 H$2$O  +  O$2$

This reaction proceeds because the reaction products are more stable than the starting compound, but this decomposition is so slow that hydrogen peroxide solutions are commercially available. In the presence of a catalyst such as manganese dioxide this reaction proceeds much more rapidly. This effect is readily seen by the effervescence of oxygen. The catalyst is not consumed in the reaction, and may be recovered unchanged and re-used indefinitely. Accordingly, manganese dioxide is said to catalyze this reaction. In living organisms, this reaction is catalyzed by enzymes (proteins that serve as catalysts) such as catalase.

Another example is the effect of catalysts on air pollution and reducing the amount of carbon monoxide. Development of active and selective catalysts for the conversion of carbon monoxide into desirable products is one of the most important roles of catalysts. Using catalysts for hydrogenation of carbon monoxide helps to remove this toxic gas and also attain useful materials.

Units
The SI derived unit for measuring the catalytic activity of a catalyst is the katal, which is quantified in moles per second. The productivity of a catalyst can be described by the turnover number (or TON) and the catalytic activity by the turn over frequency (TOF), which is the TON per time unit. The biochemical equivalent is the enzyme unit. For more information on the efficiency of enzymatic catalysis, see the article on enzymes.

Catalytic reaction mechanisms
In general, chemical reactions occur faster in the presence of a catalyst because the catalyst provides an alternative reaction mechanism (reaction pathway) having a lower activation energy than the non-catalyzed mechanism. In catalyzed mechanisms, the catalyst is regenerated.

As a simple example occurring in the gas phase, the reaction 2 SO2 + O2 → 2 SO3 can be catalyzed by adding nitric oxide. The reaction occurs in two steps:
 * 2NO + O2 → 2NO2  (rate-determining)
 * NO2 + SO2 → NO + SO3  (fast)

The NO catalyst is regenerated. The overall rate is the rate of the slow step
 * v = 2k1[NO]2[O2].

An example of heterogeneous catalysis is the reaction of oxygen and hydrogen on the surface of titanium dioxide (TiO$2$, or titania) to produce water. Scanning tunneling microscopy showed that the molecules undergo adsorption and dissociation. The dissociated, surface-bound O and H atoms diffuse together. The intermediate reaction states are: HO$2$, H$2$O$2$, then H$3$O$2$ and the reaction product (water molecule dimers), after which the water molecule desorbs from the catalyst surface.

Reaction energetics
Catalysts enable pathways that differ from the uncatalyzed reactions. These pathways have lower activation energy. Consequently, more molecular collisions have the energy needed to reach the transition state. Hence, catalysts can enable reactions that would otherwise be blocked or slowed by a kinetic barrier. The catalyst may increase the reaction rate or selectivity, or enable the reaction at lower temperatures. This effect can be illustrated with an energy profile diagram.

In the catalyzed elementary reaction, catalysts do not change the extent of a reaction: they have no effect on the chemical equilibrium of a reaction. The ratio of the forward and the reverse reaction rates is unaffected (see also thermodynamics). The second law of thermodynamics describes why a catalyst does not change the chemical equilibrium of a reaction. Suppose there was such a catalyst that shifted an equilibrium. Introducing the catalyst to the system would result in a reaction to move to the new equilibrium, producing energy. Production of energy is a necessary result since reactions are spontaneous only if Gibbs free energy is produced, and if there is no energy barrier, there is no need for a catalyst. Then, removing the catalyst would also result in a reaction, producing energy; i.e. the addition and its reverse process, removal, would both produce energy. Thus, a catalyst that could change the equilibrium would be a perpetual motion machine, a contradiction to the laws of thermodynamics. Thus, catalysts do not alter the equilibrium constant. (A catalyst can however change the equilibrium concentrations by reacting in a subsequent step. It is then consumed as the reaction proceeds, and thus it is also a reactant. Illustrative is the base-catalyzed hydrolysis of esters, where the produced carboxylic acid immediately reacts with the base catalyst and thus the reaction equilibrium is shifted towards hydrolysis.)

The catalyst stabilizes the transition state more than it stabilizes the starting material. It decreases the kinetic barrier by decreasing the difference in energy between starting material and the transition state. It does not change the energy difference between starting materials and products (thermodynamic barrier), or the available energy (this is provided by the environment as heat or light).

Related concepts
Some so-called catalysts are really precatalysts. Precatalysts convert to catalysts in the reaction. For example, Wilkinson's catalyst RhCl(PPh$3$)$3$ loses one triphenylphosphine ligand before entering the true catalytic cycle. Precatalysts are easier to store but are easily activated in situ. Because of this preactivation step, many catalytic reactions involve an induction period.

In cooperative catalysis, chemical species that improve catalytic activity are called cocatalysts or promoters.

In tandem catalysis two or more different catalysts are coupled in a one-pot reaction.

In autocatalysis, the catalyst is a product of the overall reaction, in contrast to all other types of catalysis considered in this article. The simplest example of autocatalysis is a reaction of type A + B → 2 B, in one or in several steps. The overall reaction is just A → B, so that B is a product. But since B is also a reactant, it may be present in the rate equation and affect the reaction rate. As the reaction proceeds, the concentration of B increases and can accelerate the reaction as a catalyst. In effect, the reaction accelerates itself or is autocatalyzed. An example is the hydrolysis of an ester such as aspirin to a carboxylic acid and an alcohol. In the absence of added acid catalysts, the carboxylic acid product catalyzes the hydrolysis.

Switchable catalysis refers to a type of catalysis where the catalyst can be toggled between different ground states possessing distinct reactivity, typically by applying an external stimulus. This ability to reversibly switch the catalyst allows for spatiotemporal control over catalytic activity and selectivity. The external stimuli used to switch the catalyst can include changes in temperature, pH, light, electric fields, or the addition of chemical agents.

A true catalyst can work in tandem with a sacrificial catalyst. The true catalyst is consumed in the elementary reaction and turned into a deactivated form. The sacrificial catalyst regenerates the true catalyst for another cycle. The sacrificial catalyst is consumed in the reaction, and as such, it is not really a catalyst, but a reagent. For example, osmium tetroxide (OsO4) is a good reagent for dihydroxylation, but it is highly toxic and expensive. In Upjohn dihydroxylation, the sacrificial catalyst N-methylmorpholine N-oxide (NMMO) regenerates OsO4, and only catalytic quantities of OsO4 are needed.

Classification
Catalysis may be classified as either homogeneous or heterogeneous. A homogeneous catalysis is one whose components are dispersed in the same phase (usually gaseous or liquid) as the reactant's molecules. A heterogeneous catalysis is one where the reaction components are not in the same phase. Enzymes and other biocatalysts are often considered as a third category. Similar mechanistic principles apply to heterogeneous, homogeneous, and biocatalysis.

Heterogeneous catalysis
Heterogeneous catalysts act in a different phase than the reactants. Most heterogeneous catalysts are solids that act on substrates in a liquid or gaseous reaction mixture. Important heterogeneous catalysts include zeolites, alumina, higher-order oxides, graphitic carbon, transition metal oxides, metals such as Raney nickel for hydrogenation, and vanadium(V) oxide for oxidation of sulfur dioxide into sulfur trioxide by the contact process.

Diverse mechanisms for reactions on surfaces are known, depending on how the adsorption takes place (Langmuir-Hinshelwood, Eley-Rideal, and Mars-van Krevelen). The total surface area of a solid has an important effect on the reaction rate. The smaller the catalyst particle size, the larger the surface area for a given mass of particles.

A heterogeneous catalyst has active sites, which are the atoms or crystal faces where the substrate actually binds. Active sites are atoms but are often described as a facet (edge, surface, step, etc.) of a solid. Most of the volume but also most of the surface of a heterogeneous catalyst may be catalytically inactive. Finding out the nature of the active site is technically challenging.

For example, the catalyst for the Haber process for the synthesis of ammonia from nitrogen and hydrogen is often described as iron. But detailed studies and many optimizations have led to catalysts that are mixtures of iron-potassium-calcium-aluminum-oxide. The reacting gases adsorb onto active sites on the iron particles. Once physically adsorbed, the reagents partially or wholly dissociate and form new bonds. In this way the particularly strong triple bond in nitrogen is broken, which would be extremely uncommon in the gas phase due to its high activation energy. Thus, the activation energy of the overall reaction is lowered, and the rate of reaction increases. Another place where a heterogeneous catalyst is applied is in the oxidation of sulfur dioxide on vanadium(V) oxide for the production of sulfuric acid. Many heterogeneous catalysts are in fact nanomaterials.

Heterogeneous catalysts are typically "supported," which means that the catalyst is dispersed on a second material that enhances the effectiveness or minimizes its cost. Supports prevent or minimize agglomeration and sintering of small catalyst particles, exposing more surface area, thus catalysts have a higher specific activity (per gram) on support. Sometimes the support is merely a surface on which the catalyst is spread to increase the surface area. More often, the support and the catalyst interact, affecting the catalytic reaction. Supports can also be used in nanoparticle synthesis by providing sites for individual molecules of catalyst to chemically bind. Supports are porous materials with a high surface area, most commonly alumina, zeolites or various kinds of activated carbon. Specialized supports include silicon dioxide, titanium dioxide, calcium carbonate, and barium sulfate.

Electrocatalysts
In the context of electrochemistry, specifically in fuel cell engineering, various metal-containing catalysts are used to enhance the rates of the half reactions that comprise the fuel cell. One common type of fuel cell electrocatalyst is based upon nanoparticles of platinum that are supported on slightly larger carbon particles. When in contact with one of the electrodes in a fuel cell, this platinum increases the rate of oxygen reduction either to water or to hydroxide or hydrogen peroxide.

Homogeneous catalysis
Homogeneous catalysts function in the same phase as the reactants. Typically homogeneous catalysts are dissolved in a solvent with the substrates. One example of homogeneous catalysis involves the influence of H$+$ on the esterification of carboxylic acids, such as the formation of methyl acetate from acetic acid and methanol. High-volume processes requiring a homogeneous catalyst include hydroformylation, hydrosilylation, hydrocyanation. For inorganic chemists, homogeneous catalysis is often synonymous with organometallic catalysts. Many homogeneous catalysts are however not organometallic, illustrated by the use of cobalt salts that catalyze the oxidation of p-xylene to terephthalic acid.

Organocatalysis
Whereas transition metals sometimes attract most of the attention in the study of catalysis, small organic molecules without metals can also exhibit catalytic properties, as is apparent from the fact that many enzymes lack transition metals. Typically, organic catalysts require a higher loading (amount of catalyst per unit amount of reactant, expressed in mol% amount of substance) than transition metal(-ion)-based catalysts, but these catalysts are usually commercially available in bulk, helping to lower costs. In the early 2000s, these organocatalysts were considered "new generation" and are competitive to traditional metal(-ion)-containing catalysts. Organocatalysts are supposed to operate akin to metal-free enzymes utilizing, e.g., non-covalent interactions such as hydrogen bonding. The discipline organocatalysis is divided into the application of covalent (e.g., proline, DMAP) and non-covalent (e.g., thiourea organocatalysis) organocatalysts referring to the preferred catalyst-substrate binding and interaction, respectively. The Nobel Prize in Chemistry 2021 was awarded jointly to Benjamin List and David W.C. MacMillan "for the development of asymmetric organocatalysis."

Photocatalysts
Photocatalysis is the phenomenon where the catalyst can receive light to generate an excited state that effect redox reactions. Singlet oxygen is usually produced by photocatalysis. Photocatalysts are components of dye-sensitized solar cells.

Enzymes and biocatalysts
In biology, enzymes are protein-based catalysts in metabolism and catabolism. Most biocatalysts are enzymes, but other non-protein-based classes of biomolecules also exhibit catalytic properties including ribozymes, and synthetic deoxyribozymes.

Biocatalysts can be thought of as an intermediate between homogeneous and heterogeneous catalysts, although strictly speaking soluble enzymes are homogeneous catalysts and membrane-bound enzymes are heterogeneous. Several factors affect the activity of enzymes (and other catalysts) including temperature, pH, the concentration of enzymes, substrate, and products. A particularly important reagent in enzymatic reactions is water, which is the product of many bond-forming reactions and a reactant in many bond-breaking processes.

In biocatalysis, enzymes are employed to prepare many commodity chemicals including high-fructose corn syrup and acrylamide.

Some monoclonal antibodies whose binding target is a stable molecule that resembles the transition state of a chemical reaction can function as weak catalysts for that chemical reaction by lowering its activation energy. Such catalytic antibodies are sometimes called "abzymes".

Significance

Estimates are that 90% of all commercially produced chemical products involve catalysts at some stage in the process of their manufacture. In 2005, catalytic processes generated about $900 billion in products worldwide. Catalysis is so pervasive that subareas are not readily classified. Some areas of particular concentration are surveyed below.

Energy processing
Petroleum refining makes intensive use of catalysis for alkylation, catalytic cracking (breaking long-chain hydrocarbons into smaller pieces), naphtha reforming and steam reforming (conversion of hydrocarbons into synthesis gas). Even the exhaust from the burning of fossil fuels is treated via catalysis: Catalytic converters, typically composed of platinum and rhodium, break down some of the more harmful byproducts of automobile exhaust.
 * 2 CO +  2 NO  →   2 CO$2$  +  N$2$

With regard to synthetic fuels, an old but still important process is the Fischer-Tropsch synthesis of hydrocarbons from synthesis gas, which itself is processed via water-gas shift reactions, catalyzed by iron. The Sabatier reaction produces methane from carbon dioxide and hydrogen. Biodiesel and related biofuels require processing via both inorganic and biocatalysts.

Fuel cells rely on catalysts for both the anodic and cathodic reactions.

Catalytic heaters generate flameless heat from a supply of combustible fuel.

Bulk chemicals
Some of the largest-scale chemicals are produced via catalytic oxidation, often using oxygen. Examples include nitric acid (from ammonia), sulfuric acid (from sulfur dioxide to sulfur trioxide by the contact process), terephthalic acid from p-xylene, acrylic acid from propylene or propane and acrylonitrile from propane and ammonia.

The production of ammonia is one of the largest-scale and most energy-intensive processes. In the Haber process nitrogen is combined with hydrogen over an iron oxide catalyst. Methanol is prepared from carbon monoxide or carbon dioxide but using copper-zinc catalysts.

Bulk polymers derived from ethylene and propylene are often prepared via Ziegler-Natta catalysis. Polyesters, polyamides, and isocyanates are derived via acid-base catalysis.

Most carbonylation processes require metal catalysts, examples include the Monsanto acetic acid process and hydroformylation.

Fine chemicals
Many fine chemicals are prepared via catalysis; methods include those of heavy industry as well as more specialized processes that would be prohibitively expensive on a large scale. Examples include the Heck reaction, and Friedel–Crafts reactions. Because most bioactive compounds are chiral, many pharmaceuticals are produced by enantioselective catalysis (catalytic asymmetric synthesis). (R)-1,2-Propandiol, the precursor to the antibacterial levofloxacin, can be synthesized efficiently from hydroxyacetone by using catalysts based on BINAP-ruthenium complexes, in Noyori asymmetric hydrogenation:

Food processing
One of the most obvious applications of catalysis is the hydrogenation (reaction with hydrogen gas) of fats using nickel catalyst to produce margarine. Many other foodstuffs are prepared via biocatalysis (see below).

Environment
Catalysis affects the environment by increasing the efficiency of industrial processes, but catalysis also plays a direct role in the environment. A notable example is the catalytic role of chlorine free radicals in the breakdown of ozone. These radicals are formed by the action of ultraviolet radiation on chlorofluorocarbons (CFCs).
 * Cl$·$ +  O$3$  →   ClO$·$  +  O$2$
 * ClO$·$ +  O$·$  →  Cl$·$  +  O$2$

History
The term "catalyst", broadly defined as anything that increases the rate of a process, is derived from Greek καταλύειν, meaning "to annul," or "to untie," or "to pick up". The concept of catalysis was invented by chemist Elizabeth Fulhame and described in a 1794 book, based on her novel work in oxidation–reduction reactions. The first chemical reaction in organic chemistry that knowingly used a catalyst was studied in 1811 by Gottlieb Kirchhoff, who discovered the acid-catalyzed conversion of starch to glucose. The term catalysis was later used by Jöns Jakob Berzelius in 1835 to describe reactions that are accelerated by substances that remain unchanged after the reaction. Fulhame, who predated Berzelius, did work with water as opposed to metals in her reduction experiments. Other 18th century chemists who worked in catalysis were Eilhard Mitscherlich who referred to it as contact processes, and Johann Wolfgang Döbereiner who spoke of ''contact action. ''He developed Döbereiner's lamp, a lighter based on hydrogen and a platinum sponge, which became a commercial success in the 1820s that lives on today. Humphry Davy discovered the use of platinum in catalysis. In the 1880s, Wilhelm Ostwald at Leipzig University started a systematic investigation into reactions that were catalyzed by the presence of acids and bases, and found that chemical reactions occur at finite rates and that these rates can be used to determine the strengths of acids and bases. For this work, Ostwald was awarded the 1909 Nobel Prize in Chemistry. Vladimir Ipatieff performed some of the earliest industrial scale reactions, including the discovery and commercialization of oligomerization and the development of catalysts for hydrogenation.

Inhibitors, poisons, and promoters
An added substance that lowers the rate is called a reaction inhibitor if reversible and catalyst poisons if irreversible. Promoters are substances that increase the catalytic activity, even though they are not catalysts by themselves.

Inhibitors are sometimes referred to as "negative catalysts" since they decrease the reaction rate. However the term inhibitor is preferred since they do not work by introducing a reaction path with higher activation energy; this would not lower the rate since the reaction would continue to occur by the non-catalyzed path. Instead, they act either by deactivating catalysts or by removing reaction intermediates such as free radicals. In heterogeneous catalysis, coking inhibits the catalyst, which becomes covered by polymeric side products.

The inhibitor may modify selectivity in addition to rate. For instance, in the hydrogenation of alkynes to alkenes, a palladium (Pd) catalyst partly "poisoned" with lead(II) acetate (Pb(CH$3$CO$2$)$2$) can be used (Lindlar catalyst). Without the deactivation of the catalyst, the alkene produced would be further hydrogenated to alkane.

The inhibitor can produce this effect by, e.g., selectively poisoning only certain types of active sites. Another mechanism is the modification of surface geometry. For instance, in hydrogenation operations, large planes of metal surface function as sites of hydrogenolysis catalysis while sites catalyzing hydrogenation of unsaturates are smaller. Thus, a poison that covers the surface randomly will tend to lower the number of uncontaminated large planes but leave proportionally smaller sites free, thus changing the hydrogenation vs. hydrogenolysis selectivity. Many other mechanisms are also possible.

Promoters can cover up the surface to prevent the production of a mat of coke, or even actively remove such material (e.g., rhenium on platinum in platforming). They can aid the dispersion of the catalytic material or bind to reagents.

title: Catalysis


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Circumference

[[File:Circle-withsegments.svg|thumb|

Circumference = $\pi$ × diameter = 2π × radius.]]

In geometry, the circumference (from Latin circumferens, meaning "carrying around") is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk. The is the circumference, or length, of any one of its great circles.

Circle
The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

Relationship with π
The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter $\pi.$ The first few decimal digits of the numerical value of $$\pi$$ are 3.141592653589793 ... Pi is defined as the ratio of a circle's circumference $$C$$ to its diameter $$d:$$ $$\pi = \frac{C}{d}.$$

Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference: $${C} = \pi \cdot{d} = 2\pi \cdot{r}.\!$$

The ratio of the circle's circumference to its radius is called the circle constant, and is equivalent to $$2\pi$$. The value $$2\pi$$ is also the amount of radians in one turn. The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.

In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio ($$C/d,$$ since he did not use the name π) was greater than 3$10⁄71$ but less than 3$1⁄7$ by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides. This method for approximating π was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 1040 sides.

Ellipse

Circumference is used by some authors to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse, $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$$ is $$C_{\rm{ellipse}} \sim \pi \sqrt{2\left(a^2 + b^2\right)}.$$ Some lower and upper bounds on the circumference of the canonical ellipse with $$a\geq b$$ are: $$2\pi b \leq C \leq 2\pi a,$$ $$\pi (a+b) \leq C \leq 4(a+b),$$ $$4\sqrt{a^2+b^2} \leq C \leq \pi \sqrt{2\left(a^2+b^2\right)}.$$

Here the upper bound $$2\pi a$$ is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound $$4\sqrt{a^2+b^2}$$ is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.

The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind. More precisely, $$C_{\rm{ellipse}} = 4a \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2\theta}\ d\theta,$$ where $$a$$ is the length of the semi-major axis and $$e$$ is the eccentricity $$\sqrt{1 - b^2/a^2}.$$

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title: Circumference


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Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium (also called a continuum) rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships.

Continuum mechanics treats the physical properties of solids and fluids independently of any particular coordinate system in which they are observed. These properties are represented by tensors, which are mathematical objects with the salient property of being independent of coordinate systems. This permits definition of physical properties at any point in the continuum, according to mathematically convenient continuous functions. The theories of elasticity, plasticity and fluid mechanics are based on the concepts of continuum mechanics.

Concept of a continuum
The concept of a continuum underlies the mathematical framework for studying large-scale forces and deformations in materials. Although materials are composed of discrete atoms and molecules, separated by empty space or microscopic cracks and crystallographic defects, physical phenomena can often be modeled by considering a substance distributed throughout some region of space. A continuum is a body that can be continually sub-divided into infinitesimal elements with local material properties defined at any particular point. Properties of the bulk material can therefore be described by continuous functions, and their evolution can be studied using the mathematics of calculus.

Apart from the assumption of continuity, two other independent assumptions are often employed in the study of continuum mechanics. These are homogeneity (assumption of identical properties at all locations) and isotropy (assumption of directionally invariant vector properties). If these auxiliary assumptions are not globally applicable, the material may be segregated into sections where they are applicable in order to simplify the analysis. For more complex cases, one or both of these assumptions can be dropped. In these cases, computational methods are often used to solve the differential equations describing the evolution of material properties.

Major areas
An additional area of continuum mechanics comprises elastomeric foams, which exhibit a curious hyperbolic stress-strain relationship. The elastomer is a true continuum, but a homogeneous distribution of voids gives it unusual properties.

Formulation of models
Continuum mechanics models begin by assigning a region in three-dimensional Euclidean space to the material body $$\mathcal B$$ being modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time $$t$$ is labeled $$\kappa_t(\mathcal B)$$.

A particular particle within the body in a particular configuration is characterized by a position vector

 * $$\mathbf x = \sum_{i=1}^3 x_i \mathbf e_i,$$

where $$\mathbf e_i$$ are the coordinate vectors in some frame of reference chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position $$\mathbf X$$ in some reference configuration, for example the configuration at the initial time, so that

 * $$\mathbf{x}=\kappa_t(\mathbf X).$$

This function needs to have various properties so that the model makes physical sense. $$\kappa_t(\cdot)$$ needs to be: For the mathematical formulation of the model, $$\kappa_t(\cdot)$$ is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.
 * continuous in time, so that the body changes in a way which is realistic,
 * globally invertible at all times, so that the body cannot intersect itself,
 * orientation-preserving, as transformations which produce mirror reflections are not possible in nature.

Forces in a continuum
A solid is a deformable body that possesses shear strength, sc. a solid can support shear forces (forces parallel to the material surface on which they act). Fluids, on the other hand, do not sustain shear forces.

Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces $$\mathbf F_C$$ and body forces $$\mathbf F_B$$. Thus, the total force $$\mathcal F$$ applied to a body or to a portion of the body can be expressed as:

 * $$\mathcal F = \mathbf F_C + \mathbf F_B$$

Surface forces
Surface forces or contact forces, expressed as force per unit area, can act either on the bounding surface of the body, as a result of mechanical contact with other bodies, or on imaginary internal surfaces that bound portions of the body, as a result of the mechanical interaction between the parts of the body to either side of the surface (Euler-Cauchy's stress principle). When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to Newton's third law of motion of conservation of linear momentum and angular momentum (for continuous bodies these laws are called the Euler's equations of motion). The internal contact forces are related to the body's deformation through constitutive equations. The internal contact forces may be mathematically described by how they relate to the motion of the body, independent of the body's material makeup.

The distribution of internal contact forces throughout the volume of the body is assumed to be continuous. Therefore, there exists a contact force density or Cauchy traction field $$\mathbf T(\mathbf n, \mathbf x, t)$$ that represents this distribution in a particular configuration of the body at a given time $$t\,\!$$. It is not a vector field because it depends not only on the position $$\mathbf x$$ of a particular material point, but also on the local orientation of the surface element as defined by its normal vector $$\mathbf n$$.

Any differential area $$dS\,\!$$ with normal vector $$\mathbf n$$ of a given internal surface area $$S\,\!$$, bounding a portion of the body, experiences a contact force $$d\mathbf F_C\,\!$$ arising from the contact between both portions of the body on each side of $$S\,\!$$, and it is given by

 * $$d\mathbf F_C= \mathbf T^{(\mathbf n)}\,dS$$

where $$\mathbf T^{(\mathbf n)}$$ is the surface traction, also called stress vector, traction, or traction vector. The stress vector is a frame-indifferent vector (see Euler-Cauchy's stress principle).

The total contact force on the particular internal surface $$S\,\!$$ is then expressed as the sum (surface integral) of the contact forces on all differential surfaces $$dS\,\!$$:

 * $$\mathbf F_C=\int_S \mathbf T^{(\mathbf n)}\,dS$$

In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces (ionic, metallic, and van der Waals forces) required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction. Stresses generated during manufacture of the body to a specific configuration are also excluded when considering stresses in a body. Therefore, the stresses considered in continuum mechanics are only those produced by deformation of the body, sc. only relative changes in stress are considered, not the absolute values of stress.

Body forces
Body forces are forces originating from sources outside of the body that act on the volume (or mass) of the body. Saying that body forces are due to outside sources implies that the interaction between different parts of the body (internal forces) are manifested through the contact forces alone. These forces arise from the presence of the body in force fields, e.g. gravitational field (gravitational forces) or electromagnetic field (electromagnetic forces), or from inertial forces when bodies are in motion. As the mass of a continuous body is assumed to be continuously distributed, any force originating from the mass is also continuously distributed. Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i.e. acting on every point in it. Body forces are represented by a body force density $$\mathbf b(\mathbf x, t)$$ (per unit of mass), which is a frame-indifferent vector field.

In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density $$\mathbf \rho (\mathbf x, t)\,\!$$ of the material, and it is specified in terms of force per unit mass ($$b_i\,\!$$) or per unit volume ($$p_i\,\!$$). These two specifications are related through the material density by the equation $$\rho b_i = p_i\,\!$$. Similarly, the intensity of electromagnetic forces depends upon the strength (electric charge) of the electromagnetic field.

The total body force applied to a continuous body is expressed as

 * $$\mathbf F_B=\int_V\mathbf b\,dm=\int_V \rho\mathbf b\,dV$$

Body forces and contact forces acting on the body lead to corresponding moments of force (torques) relative to a given point. Thus, the total applied torque $$\mathcal M$$ about the origin is given by

 * $$\mathcal M= \mathbf M_C + \mathbf M_B$$

In certain situations, not commonly considered in the analysis of the mechanical behavior of materials, it becomes necessary to include two other types of forces: these are couple stresses (surface couples, contact torques) and body moments. Couple stresses are moments per unit area applied on a surface. Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. Both are important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration (e.g. bones), solids under the action of an external magnetic field, and the dislocation theory of metals.

Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called polar materials. Non-polar materials are then those materials with only moments of forces. In the classical branches of continuum mechanics the development of the theory of stresses is based on non-polar materials.

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) in the body can be given by

 * $$\mathcal F = \int_V \mathbf a\,dm = \int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV$$
 * $$\mathcal M = \int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV$$

Kinematics: motion and deformation
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration $$\kappa_0(\mathcal B)$$ to a current or deformed configuration $$\kappa_t(\mathcal B)$$ (Figure 2).

The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a path line.

There is continuity during motion or deformation of a continuum body in the sense that:
 * The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
 * The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at $$t=0$$ is considered the reference configuration, $$\kappa_0 (\mathcal B)$$. The components $$X_i$$ of the position vector $$\mathbf X$$ of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.

When analyzing the motion or deformation of solids, or the flow of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

Lagrangian description
In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case the reference configuration is the configuration at $$t=0$$. An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, $$\kappa_0(\mathcal B)$$. This description is normally used in solid mechanics.

In the Lagrangian description, the motion of a continuum body is expressed by the mapping function $$\chi(\cdot)$$ (Figure 2),

 * $$\mathbf x=\chi(\mathbf X, t)$$

which is a mapping of the initial configuration $$\kappa_0(\mathcal B)$$ onto the current configuration $$\kappa_t(\mathcal B)$$, giving a geometrical correspondence between them, i.e. giving the position vector $$\mathbf{x}=x_i\mathbf e_i$$ that a particle $$X$$, with a position vector $$\mathbf X$$ in the undeformed or reference configuration $$\kappa_0(\mathcal B)$$, will occupy in the current or deformed configuration $$\kappa_t(\mathcal B)$$ at time $$t$$. The components $$x_i$$ are called the spatial coordinates.

Physical and kinematic properties $$P_{ij\ldots}$$, i.e. thermodynamic properties and flow velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. $$P_{ij\ldots}=P_{ij\ldots}(\mathbf X,t)$$.

The material derivative of any property $$P_{ij\ldots}$$ of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body. The material derivative is also known as the substantial derivative, or comoving derivative, or convective derivative. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles.

In the Lagrangian description, the material derivative of $$P_{ij\ldots}$$ is simply the partial derivative with respect to time, and the position vector $$\mathbf X$$ is held constant as it does not change with time. Thus, we have

 * $$\frac{d}{dt}[P_{ij\ldots}(\mathbf X,t)]=\frac{\partial}{\partial t}[P_{ij\ldots}(\mathbf X,t)]$$

The instantaneous position $$\mathbf x$$ is a property of a particle, and its material derivative is the instantaneous flow velocity $$\mathbf v$$ of the particle. Therefore, the flow velocity field of the continuum is given by

 * $$\mathbf v = \dot{\mathbf x} =\frac{d\mathbf x}{dt}=\frac{\partial \chi(\mathbf X,t)}{\partial t} $$

Similarly, the acceleration field is given by

 * $$\mathbf a= \dot{\mathbf v} = \ddot{\mathbf x} =\frac{d^2\mathbf x}{dt^2}=\frac{\partial^2 \chi(\mathbf X,t)}{\partial t^2} $$

Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points. All physical quantities characterizing the continuum are described this way. In this sense, the function $$\chi(\cdot)$$ and $$P_{ij\ldots}(\cdot)$$ are single-valued and continuous, with continuous derivatives with respect to space and time to whatever order is required, usually to the second or third.

Eulerian description
Continuity allows for the inverse of $$\chi(\cdot)$$ to trace backwards where the particle currently located at $$\mathbf x$$ was located in the initial or referenced configuration $$\kappa_0(\mathcal B)$$. In this case the description of motion is made in terms of the spatial coordinates, in which case is called the spatial description or Eulerian description, i.e. the current configuration is taken as the reference configuration.

The Eulerian description, introduced by d'Alembert, focuses on the current configuration $$\kappa_t(\mathcal B)$$, giving attention to what is occurring at a fixed point in space as time progresses, instead of giving attention to individual particles as they move through space and time. This approach is conveniently applied in the study of fluid flow where the kinematic property of greatest interest is the rate at which change is taking place rather than the shape of the body of fluid at a reference time.

Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function

 * $$\mathbf X=\chi^{-1}(\mathbf x, t)$$

which provides a tracing of the particle which now occupies the position $$\mathbf x$$ in the current configuration $$\kappa_t(\mathcal B)$$ to its original position $$\mathbf X$$ in the initial configuration $$\kappa_0(\mathcal B)$$.

A necessary and sufficient condition for this inverse function to exist is that the determinant of the Jacobian matrix, often referred to simply as the Jacobian, should be different from zero. Thus,

 * $$J = \left| \frac{\partial \chi_i}{\partial X_J} \right| = \left| \frac{\partial x_i}{\partial X_J} \right| \neq 0$$

In the Eulerian description, the physical properties $$P_{ij\ldots}$$ are expressed as

 * $$P_{ij \ldots}=P_{ij\ldots}(\mathbf X,t)=P_{ij\ldots}[\chi^{-1}(\mathbf x,t),t]=p_{ij\ldots}(\mathbf x,t)$$

where the functional form of $$P_{ij \ldots}$$ in the Lagrangian description is not the same as the form of $$p_{ij \ldots}$$ in the Eulerian description.

The material derivative of $$p_{ij\ldots}(\mathbf x,t)$$, using the chain rule, is then

 * $$\frac{d}{dt}[p_{ij\ldots}(\mathbf x,t)]=\frac{\partial}{\partial t}[p_{ij\ldots}(\mathbf x,t)]+ \frac{\partial}{\partial x_k}[p_{ij\ldots}(\mathbf x,t)]\frac{dx_k}{dt}$$

The first term on the right-hand side of this equation gives the local rate of change of the property $$p_{ij\ldots}(\mathbf x,t)$$ occurring at position $$\mathbf x$$. The second term of the right-hand side is the convective rate of change and expresses the contribution of the particle changing position in space (motion).

Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the flow velocity field. All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position $$\mathbf x$$.

Displacement field
The vector joining the positions of a particle $$P$$ in the undeformed configuration and deformed configuration is called the displacement vector $$\mathbf u(\mathbf X,t)=u_i\mathbf e_i$$, in the Lagrangian description, or $$\mathbf U(\mathbf x,t)=U_J\mathbf E_J$$, in the Eulerian description.

A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field, In general, the displacement field is expressed in terms of the material coordinates as

 * $$\mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J$$

or in terms of the spatial coordinates as

 * $$\mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,$$

where $$\alpha_{Ji}$$ are the direction cosines between the material and spatial coordinate systems with unit vectors $$\mathbf E_J$$ and $$\mathbf e_i$$, respectively. Thus

 * $$\mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}$$

and the relationship between $$u_i$$ and $$U_J$$ is then given by

 * $$u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i$$

Knowing that
 * $$\mathbf e_i = \alpha_{iJ}\mathbf E_J$$

then
 * $$\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)$$

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in $$\mathbf b=0$$, and the direction cosines become Kronecker deltas, i.e.

 * $$\mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}$$

Thus, we have

 * $$\mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J$$

or in terms of the spatial coordinates as

 * $$\mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J $$

Governing equations
Continuum mechanics deals with the behavior of materials that can be approximated as continuous for certain length and time scales. The equations that govern the mechanics of such materials include the balance laws for mass, momentum, and energy. Kinematic relations and constitutive equations are needed to complete the system of governing equations. Physical restrictions on the form of the constitutive relations can be applied by requiring that the second law of thermodynamics be satisfied under all conditions. In the continuum mechanics of solids, the second law of thermodynamics is satisfied if the Clausius–Duhem form of the entropy inequality is satisfied.

The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes:

 * 1) the physical quantity itself flows through the surface that bounds the volume,
 * 2) there is a source of the physical quantity on the surface of the volume, or/and,
 * 3) there is a source of the physical quantity inside the volume.

Let $$\Omega$$ be the body (an open subset of Euclidean space) and let  $$\partial \Omega $$ be its surface (the boundary of $$\Omega$$).

Let the motion of material points in the body be described by the map
 * $$\mathbf{x} = \boldsymbol{\chi}(\mathbf{X}) = \mathbf{x}(\mathbf{X})$$

where $$\mathbf{X}$$ is the position of a point in the initial configuration and $$\mathbf{x}$$ is the location of the same point in the deformed configuration.

The deformation gradient is given by
 * $$\boldsymbol{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \nabla \mathbf{x} ~.$$

Balance laws
Let $$f(\mathbf{x},t)$$ be a physical quantity that is flowing through the body. Let $$g(\mathbf{x},t)$$ be sources on the surface of the body and let $$h(\mathbf{x},t)$$ be sources inside the body. Let $$\mathbf{n}(\mathbf{x},t)$$ be the outward unit normal to the surface $$\partial \Omega $$. Let $$\mathbf{v}(\mathbf{x},t)$$ be the flow velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface $$\partial \Omega $$ is moving be $$u_n$$ (in the direction $$\mathbf{n}$$).

Then, balance laws can be expressed in the general form

\cfrac{d}{dt}\left[\int_{\Omega} f(\mathbf{x},t)~\text{dV}\right] = \int_{\partial \Omega } f(\mathbf{x},t)[u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)]~\text{dA} + \int_{\partial \Omega } g(\mathbf{x},t)~\text{dA} + \int_{\Omega} h(\mathbf{x},t)~\text{dV} ~. $$ The functions $$f(\mathbf{x},t)$$, $$g(\mathbf{x},t)$$, and $$h(\mathbf{x},t)$$ can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with. If there are internal boundaries in the body, jump discontinuities also need to be specified in the balance laws.

If we take the Eulerian point of view, it can be shown that the balance laws of mass, momentum, and energy for a solid can be written as (assuming the source term is zero for the mass and angular momentum equations)

{   \begin{align} \dot{\rho} + \rho (\boldsymbol{\nabla} \cdot \mathbf{v}) & = 0 & & \qquad\text{Balance of Mass} \\ \rho~\dot{\mathbf{v}} - \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} - \rho~\mathbf{b} & = 0 & & \qquad\text{Balance of Linear Momentum (Cauchy's first law of motion)} \\ \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T & & \qquad\text{Balance of Angular Momentum (Cauchy's second law of motion)} \\ \rho~\dot{e} - \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s & = 0 & & \qquad\text{Balance of Energy.} \end{align} } $$

In the above equations $$\rho(\mathbf{x},t)$$ is the mass density (current),  $$\dot{\rho}$$ is the material time derivative of $$\rho$$, $$\mathbf{v}(\mathbf{x},t)$$ is the particle velocity, $$\dot{\mathbf{v}}$$ is the material time derivative of $$\mathbf{v}$$, $$\boldsymbol{\sigma}(\mathbf{x},t)$$ is the Cauchy stress tensor, $$\mathbf{b}(\mathbf{x},t)$$ is the body force density, $$e(\mathbf{x},t)$$ is the internal energy per unit mass, $$\dot{e}$$ is the material time derivative of $$e$$, $$\mathbf{q}(\mathbf{x},t)$$ is the heat flux vector, and $$s(\mathbf{x},t)$$ is an energy source per unit mass. The operators used are defined below. With respect to the reference configuration (the Lagrangian point of view), the balance laws can be written as

{   \begin{align} \rho~\det(\boldsymbol{F}) - \rho_0 &= 0 & & \qquad \text{Balance of Mass} \\ \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{P}^T -\rho_0~\mathbf{b} & = 0 & & \qquad \text{Balance of Linear Momentum} \\ \boldsymbol{F}\cdot\boldsymbol{P}^T & = \boldsymbol{P}\cdot\boldsymbol{F}^T & & \qquad \text{Balance of Angular Momentum} \\ \rho_0~\dot{e} - \boldsymbol{P}^T:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0 & & \qquad\text{Balance of Energy.} \end{align} } $$

In the above, $$\boldsymbol{P}$$ is the first Piola-Kirchhoff stress tensor, and  $$\rho_0$$ is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by

\boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T} ~\text{where}~ J = \det(\boldsymbol{F}) $$

We can alternatively define the nominal stress tensor $$\boldsymbol{N}$$ which is the transpose of the first Piola-Kirchhoff stress tensor such that

\boldsymbol{N} = \boldsymbol{P}^T = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} ~. $$ Then the balance laws become

{   \begin{align} \rho~\det(\boldsymbol{F}) - \rho_0 &= 0 & & \qquad \text{Balance of Mass} \\ \rho_0~\ddot{\mathbf{x}} - \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{N} -\rho_0~\mathbf{b} & = 0 & & \qquad \text{Balance of Linear Momentum} \\ \boldsymbol{F}\cdot\boldsymbol{N} & = \boldsymbol{N}^T\cdot\boldsymbol{F}^T & & \qquad \text{Balance of Angular Momentum} \\ \rho_0~\dot{e} - \boldsymbol{N}:\dot{\boldsymbol{F}} + \boldsymbol{\nabla}_{\circ}\cdot\mathbf{q} - \rho_0~s & = 0 & & \qquad\text{Balance of Energy.} \end{align} } $$

Operators
The operators in the above equations are defined as

\boldsymbol{\nabla} \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial x_j}\mathbf{e}_i\otimes\mathbf{e}_j = v_{i,j}\mathbf{e}_i\otimes\mathbf{e}_j ~; \boldsymbol{\nabla} \cdot \mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial x_i} = v_{i,i} ~; \boldsymbol{\nabla} \cdot \boldsymbol{S} = \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial x_j}~\mathbf{e}_i = \sigma_{ij,j}~\mathbf{e}_i ~. $$ where $$\mathbf{v}$$ is a vector field, $$\boldsymbol{S}$$ is a second-order tensor field, and $$\mathbf{e}_i$$ are the components of an orthonormal basis in the current configuration. Also,

\boldsymbol{\nabla}_{\circ} \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial X_j}\mathbf{E}_i\otimes\mathbf{E}_j = v_{i,j}\mathbf{E}_i\otimes\mathbf{E}_j ~; \boldsymbol{\nabla}_{\circ}\cdot\mathbf{v} = \sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~; \boldsymbol{\nabla}_{\circ}\cdot\boldsymbol{S} = \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial X_j}~\mathbf{E}_i = S_{ij,j}~\mathbf{E}_i $$ where $$\mathbf{v}$$ is a vector field, $$\boldsymbol{S}$$ is a second-order tensor field, and $$\mathbf{E}_i$$ are the components of an orthonormal basis in the reference configuration.

The inner product is defined as

\boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = \operatorname{trace}(\boldsymbol{A}\boldsymbol{B}^T) ~. $$

Clausius–Duhem inequality
The Clausius–Duhem inequality can be used to express the second law of thermodynamics for elastic-plastic materials. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.

Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass. The quantity of interest in this case is the entropy. Thus, we assume that there is an entropy flux, an entropy source, an internal mass density $$\rho$$ and an internal specific entropy (i.e. entropy per unit mass) $$\eta$$ in the region of interest.

Let $$\Omega$$ be such a region and let $$\partial \Omega $$ be its boundary. Then the second law of thermodynamics states that the rate of increase of $$\eta$$ in this region is greater than or equal to the sum of that supplied to $$\Omega$$ (as a flux or from internal sources) and the change of the internal entropy density $$\rho\eta$$ due to material flowing in and out of the region. Let $$\partial \Omega $$ move with a flow velocity $$u_n$$ and let particles inside $$\Omega$$ have velocities $$\mathbf{v}$$. Let $$\mathbf{n}$$ be the unit outward normal to the surface $$\partial \Omega $$. Let $$\rho$$ be the density of matter in the region, $$\bar{q}$$ be the entropy flux at the surface, and $$r$$ be the entropy source per unit mass. Then the entropy inequality may be written as

\cfrac{d}{dt}\left(\int_{\Omega} \rho~\eta~\text{dV}\right) \ge \int_{\partial \Omega} \rho~\eta~(u_n - \mathbf{v}\cdot\mathbf{n}) ~\text{dA} + \int_{\partial \Omega} \bar{q}~\text{dA} + \int_{\Omega} \rho~r~\text{dV}. $$ The scalar entropy flux can be related to the vector flux at the surface by the relation $$\bar{q} = -\boldsymbol{\psi}(\mathbf{x})\cdot\mathbf{n}$$. Under the assumption of incrementally isothermal conditions, we have

\boldsymbol{\psi}(\mathbf{x}) = \cfrac{\mathbf{q}(\mathbf{x})}{T} ~; r = \cfrac{s}{T} $$ where $$\mathbf{q}$$ is the heat flux vector, $$s$$ is an energy source per unit mass, and $$T$$ is the absolute temperature of a material point at $$\mathbf{x}$$ at time $$t$$.

We then have the Clausius–Duhem inequality in integral form:

{   \cfrac{d}{dt}\left(\int_{\Omega} \rho~\eta~\text{dV}\right) \ge \int_{\partial \Omega} \rho~\eta~(u_n - \mathbf{v}\cdot\mathbf{n}) ~\text{dA} - \int_{\partial \Omega} \cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~\text{dA} + \int_\Omega \cfrac{\rho~s}{T}~\text{dV}. } $$ We can show that the entropy inequality may be written in differential form as

{   \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) + \cfrac{\rho~s}{T}. } $$ In terms of the Cauchy stress and the internal energy, the Clausius–Duhem inequality may be written as

{     \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}. } $$

Validity
The validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exist. More specifically, the continuum hypothesis hinges on the concepts of a representative elementary volume and separation of scales based on the Hill–Mandel condition. This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure. When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than the size of the representative volume element (RVE), a statistical volume element (SVE) is employed, which results in random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. Experimentally, the RVE can only be evaluated when the constitutive response is spatially homogenous.

Applications

 * Continuum mechanics
 * Solid mechanics
 * Fluid mechanics
 * Engineering
 * Civil engineering
 * Mechanical engineering
 * Aerospace engineering
 * Biomedical engineering
 * Chemical engineering

General references

title: Continuum mechanics


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Constitutional law

Constitutional law is a body of law which defines the role, powers, and structure of different entities within a state, namely, the executive, the parliament or legislature, and the judiciary; as well as the basic rights of citizens and, in federal countries such as the United States and Canada, the relationship between the central government and state, provincial, or territorial governments.

Not all nation states have codified constitutions, though all such states have a jus commune, or law of the land, that may consist of a variety of imperative and consensual rules. These may include customary law, conventions, statutory law, judge-made law, or international rules and norms. Constitutional law deals with the fundamental principles by which the government exercises its authority. In some instances, these principles grant specific powers to the government, such as the power to tax and spend for the welfare of the population. Other times, constitutional principles act to place limits on what the government can do, such as prohibiting the arrest of an individual without sufficient cause.

In most nations, such as the United States, India, and Singapore, constitutional law is based on the text of a document ratified at the time the nation came into being. Other constitutions, notably that of the United Kingdom, rely heavily on uncodified rules, as several legislative statutes and constitutional conventions, their status within constitutional law varies, and the terms of conventions are in some cases strongly contested.

State and legal structure
Constitutional laws can be considered second order rule making or rules about making rules to exercise power. It governs the relationships between the judiciary, the legislature and the executive with the bodies under its authority. One of the key tasks of constitutions within this context is to indicate hierarchies and relationships of power. For example, in a unitary state, the constitution will vest ultimate authority in one central administration and legislature, and judiciary, though there is often a delegation of power or authority to local or municipal authorities. When a constitution establishes a federal state, it will identify multiple levels of government coexisting with exclusive or shared areas of jurisdiction over lawmaking, application and enforcement. Some federal states, most notably the United States, have separate and parallel federal and state judiciaries, with each having its own hierarchy of courts with a supreme court for each state. India, on the other hand, has one judiciary divided into district courts, high courts, and the Supreme Court of India.

Human rights
Human rights or civil liberties form a crucial part of a country's constitution and uphold the rights of the individual against the state. Most jurisdictions, like the United States and France, have a codified constitution, with a bill of rights. A recent example is the Charter of Fundamental Rights of the European Union which was intended to be included in the Treaty establishing a Constitution for Europe, that failed to be ratified. Perhaps the most important example is the Universal Declaration of Human Rights under the UN Charter. These are intended to ensure basic political, social and economic standards that a nation state, or intergovernmental body is obliged to provide to its citizens but many do include its governments. Canada is another instance where a codified constitution. with the Canadian Charter of Rights and Freedoms, protects human rights for people under the nation's jurisdiction.

Some countries like the United Kingdom have no entrenched document setting out fundamental rights; in those jurisdictions the constitution is composed of statute, case law and convention. A case named Entick v. Carrington is a constitutional principle deriving from the common law. John Entick's house was searched and ransacked by Sherriff Carrington. Carrington argued that a warrant from a Government minister, the Earl of Halifax was valid authority, even though there was no statutory provision or court order for it. The court, led by Lord Camden stated that,

"'The great end, for which men entered into society, was to secure their property. That right is preserved sacred and incommunicable in all instances, where it has not been taken away or abridged by some public law for the good of the whole. By the laws of England, every invasion of private property, be it ever so minute, is a trespass... If no excuse can be found or produced, the silence of the books is an authority against the defendant, and the plaintiff must have judgment.'"

The common law and the civil law jurisdictions do not share the same constitutional law underpinnings. Common law nations, such as those in the Commonwealth as well as the United States, derive their legal systems from that of the United Kingdom, and as such place emphasis on judicial precedent, whereby consequential court rulings (especially those by higher courts) are a source of law. Civil law jurisdictions, on the other hand, place less emphasis on judicial review and only the parliament or legislature has the power to effect law. As a result, the structure of the judiciary differs significantly between the two, with common law judiciaries being adversarial and civil law judiciaries being inquisitorial. Common law judicatures consequently separate the judiciary from the prosecution,  thereby establishing the courts as completely independent from both the legislature and law enforcement. Human rights law in these countries is as a result, largely built on legal precedent in the courts' interpretation of constitutional law, whereas that of civil law countries is almost exclusively composed of codified law, constitutional or otherwise.

Legislative procedure
Another main function of constitutions may be to describe the procedure by which parliaments may legislate. For instance, special majorities may be required to alter the constitution. In bicameral legislatures, there may be a process laid out for second or third readings of bills before a new law can enter into force. Alternatively, there may further be requirements for maximum terms that a government can keep power before holding an election.

Study of constitutional law
Constitutional law is a major focus of legal studies and research. For example, most law students in the United States are required to take a class in Constitutional Law during their first year, and several law journals are devoted to the discussion of constitutional issues.

The rule of law
The doctrine of the rule of law dictates that government must be conducted according to law. This was first established by British legal theorist A. V. Dicey.

Dicey identified three essential elements of the British Constitution which were indicative of the rule of law:
 * 1) Absolute supremacy of regular law as opposed to the influence of arbitrary power;
 * 2) Equality before the law;
 * 3) The Constitution is a result of the ordinary law of the land.

Dicey's rule of law formula consists of three classic tenets. The first is that the regular law is supreme over arbitrary and discretionary powers. "[N]o man is punishable ... except for a distinct breach of the law established in the ordinary legal manner before the ordinary courts of the land."

The second is that all men are to stand equal in the eyes of the law. "...no man is above the law...every man, whatever be his rank or condition, is subject to the ordinary law of the realm and amenable to the jurisdiction of the ordinary tribunals"

The third is that the general ideas and principles that the constitution supports arise directly from the judgements and precedents issued by the judiciary. "We may say that the constitution is pervaded by the rule of law on the ground that the general principles of the constitution... are with us the result of judicial decisions determining the rights of private persons in particular cases brought before the courts"

The separation of powers
Separation of powers is often regarded as a second limb functioning alongside the rule of law to curb the powers of the government. In many modern nation states, power is divided and vested into three branches of government: The legislature, the executive, and the judiciary are known as the horizontal separation of powers. The first and the second are harmonised in traditional Westminster system. Vertical separation of powers is decentralisation.

Election law
Election law is a subfield of constitutional law. It includes the rules governing the process of elections. These rules enable the translation of the will of the people into functioning democracies. Election law addresses issues who is entitled to vote, voter registration, ballot access, campaign finance and party funding, redistricting, apportionment, electronic voting and voting machines, accessibility of elections, election systems and formulas, vote counting, election disputes, referendums, and issues such as electoral fraud and electoral silence.

title: Constitutional law


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Celtic languages

The Celtic languages are a branch of the Indo-European language family, descended from Proto-Celtic. The term "Celtic" was first used to describe this language group by Edward Lhuyd in 1707, following Paul-Yves Pezron, who made the explicit link between the Celts described by classical writers and the Welsh and Breton languages.

During the first millennium BC, Celtic languages were spoken across much of Europe and central Anatolia. Today, they are restricted to the northwestern fringe of Europe and a few diaspora communities. There are six living languages: the four continuously living languages Breton, Irish, Scottish Gaelic and Welsh, and the two revived languages Cornish and Manx. All are minority languages in their respective countries, though there are continuing efforts at revitalisation. Welsh is an official language in Wales and Irish is an official language of Ireland and of the European Union. Welsh is the only Celtic language not classified as endangered by UNESCO. The Cornish and Manx languages became extinct in modern times but have been revived. Each now has several hundred second-language speakers.

Irish, Manx and Scottish Gaelic form the Goidelic languages, while Welsh, Cornish and Breton are Brittonic. All of these are Insular Celtic languages, since Breton, the only living Celtic language spoken in continental Europe, is descended from the language of settlers from Britain. There are a number of extinct but attested continental Celtic languages, such as Celtiberian, Galatian and Gaulish. Beyond that there is no agreement on the subdivisions of the Celtic language family. They may be divided into P-Celtic and Q-Celtic.

The Celtic languages have a rich literary tradition. The earliest specimens of written Celtic are Lepontic inscriptions from the 6th century BC in the Alps. Early Continental inscriptions used Italic and Paleohispanic scripts. Between the 4th and 8th centuries, Irish and Pictish were occasionally written in an original script, Ogham, but Latin script came to be used for all Celtic languages. Welsh has had a continuous literary tradition from the 6th century AD.

Living languages
SIL Ethnologue lists six living Celtic languages, of which four have retained a substantial number of native speakers. These are: the Goidelic languages (Irish and Scottish Gaelic, both descended from Middle Irish) and the Brittonic languages (Welsh and Breton, descended from Common Brittonic). The other two, Cornish (Brittonic) and Manx (Goidelic), died out in modern times with their presumed last native speakers in 1777 and 1974 respectively. Revitalisation movements in the 2000s led to the reemergence of native speakers for both languages following their adoption by adults and children. By the 21st century, there were roughly one million native speakers of Celtic languages, increasing to 1.4 million speakers by 2010.

Mixed languages

 * Beurla Reagaird, Highland travellers' language
 * Shelta, based largely on Irish and Hiberno-English (some 86,000 speakers in 2009).

Classification

Celtic is divided into various branches:

 * Lepontic, the oldest attested Celtic language (from the 6th century BC). Anciently spoken in Switzerland and in Northern-Central Italy. Coins with Lepontic inscriptions have been found in Noricum and Gallia Narbonensis.
 * Celtiberian, also called Eastern or Northeastern Hispano-Celtic, spoken in the ancient Iberian Peninsula, in the eastern part of Old Castile and south of Aragon. Modern provinces: Segovia, Burgos, Soria, Guadalajara, Cuenca, Zaragoza and Teruel. The relationship of Celtiberian with Gallaecian, in northwest Iberia, is uncertain.
 * Gallaecian, also known as Western or Northwestern Hispano-Celtic, anciently spoken in the northwest of the peninsula (modern Northern Portugal, and the Spanish regions of Galicia, Asturias and northwestern Castile and León).
 * Gaulish languages, including Galatian and possibly Noric. These were once spoken in a wide arc from Belgium to Turkey. They are now all extinct.
 * Brittonic, spoken in Great Britain and Brittany. Including the living languages Breton, Cornish, and Welsh, and the lost Cumbric and Pictish, though Pictish may be a sister language rather than a daughter of Common Brittonic. Before the arrival of Scotti on the Isle of Man in the 9th century, there may have been a Brittonic language there. The theory of a Brittonic Ivernic language predating Goidelic speech in Ireland has been suggested, but is not widely accepted.
 * Goidelic, including the extant Irish, Manx, and Scottish Gaelic.

Continental/Insular Celtic and P/Q-Celtic hypotheses
Scholarly handling of Celtic languages has been contentious owing to scarceness of primary source data. Some scholars (such as Cowgill 1975; McCone 1991, 1992; and Schrijver 1995) posit that the primary distinction is between Continental Celtic and Insular Celtic, arguing that the differences between the Goidelic and Brittonic languages arose after these split off from the Continental Celtic languages. Other scholars (such as Schmidt 1988) make the primary distinction between P-Celtic and Q-Celtic languages based on the replacement of initial Q by initial P in some words. Most of the Gallic and Brittonic languages are P-Celtic, while the Goidelic and Hispano-Celtic (or Celtiberian) languages are Q-Celtic. The P-Celtic languages (also called Gallo-Brittonic) are sometimes seen (for example by Koch 1992) as a central innovating area as opposed to the more conservative peripheral Q-Celtic languages. According to Ranko Matasovic in the introduction to his 2009 Etymological Dictionary of Proto-Celtic : "Celtiberian...is almost certainly an independent branch on the Celtic genealogical tree, one that became separated from the others very early."

The Breton language is Brittonic, not Gaulish, though there may be some input from the latter, having been introduced from Southwestern regions of Britain in the post-Roman era and having evolved into Breton.

In the P/Q classification schema, the first language to split off from Proto-Celtic was Gaelic. It has characteristics that some scholars see as archaic, but others see as also being in the Brittonic languages (see Schmidt). In the Insular/Continental classification schema, the split of the former into Gaelic and Brittonic is seen as being late.

The distinction of Celtic into these four sub-families most likely occurred about 900 BC according to Gray and Atkinson but, because of estimation uncertainty, it could be any time between 1200 and 800 BC. However, they only considered Gaelic and Brythonic. The controversial paper by Forster and Toth included Gaulish and put the break-up much earlier at 3200 BC ± 1500 years. They support the Insular Celtic hypothesis. The early Celts were commonly associated with the archaeological Urnfield culture, the Hallstatt culture, and the La Tène culture, though the earlier assumption of association between language and culture is now considered to be less strong.

There are legitimate scholarly arguments for both the Insular Celtic hypothesis and the P-/Q-Celtic hypothesis. Proponents of each schema dispute the accuracy and usefulness of the other's categories. However, since the 1970s the division into Insular and Continental Celtic has become the more widely held view (Cowgill 1975; McCone 1991, 1992; Schrijver 1995), but in the middle of the 1980s, the P-/Q-Celtic theory found new supporters (Lambert 1994), because of the inscription on the Larzac piece of lead (1983), the analysis of which reveals another common phonetical innovation -nm- > -nu (Gaelic ainm / Gaulish anuana, Old Welsh enuein "names"), that is less accidental than only one. The discovery of a third common innovation would allow the specialists to come to the conclusion of a Gallo-Brittonic dialect (Schmidt 1986; Fleuriot 1986).

The interpretation of this and further evidence is still quite contested, and the main argument for Insular Celtic is connected with the development of verbal morphology and the syntax in Irish and British Celtic, which Schumacher regards as convincing, while he considers the P-Celtic/Q-Celtic division unimportant and treats Gallo-Brittonic as an outdated theory. Stifter affirms that the Gallo-Brittonic view is "out of favour" in the scholarly community as of 2008 and the Insular Celtic hypothesis "widely accepted".

When referring only to the modern Celtic languages, since no Continental Celtic language has living descendants, "Q-Celtic" is equivalent to "Goidelic" and "P-Celtic" is equivalent to "Brittonic".

How the family tree of the Celtic languages is ordered depends on which hypothesis is used:

"Insular Celtic hypothesis"

 * Proto-Celtic
 * Continental Celtic
 * Celtiberian
 * Gallaecian
 * Gaulish
 * Insular Celtic
 * Brittonic
 * Goidelic

"P/Q-Celtic hypothesis"

 * Proto-Celtic
 * Q-Celtic
 * Celtiberian
 * Gallaecian
 * Goidelic
 * P-Celtic
 * Gaulish
 * Brittonic

Eska (2010)
Eska evaluates the evidence as supporting the following tree, based on shared innovations, though it is not always clear that the innovations are not areal features. It seems likely that Celtiberian split off before Cisalpine Celtic, but the evidence for this is not robust. On the other hand, the unity of Gaulish, Goidelic, and Brittonic is reasonably secure. Schumacher (2004, p. 86) had already cautiously considered this grouping to be likely genetic, based, among others, on the shared reformation of the sentence-initial, fully inflecting relative pronoun *i̯os, *i̯ā, *i̯od into an uninflected enclitic particle. Eska sees Cisalpine Gaulish as more akin to Lepontic than to Transalpine Gaulish.

 * Celtic
 * Hispano-Celtic
 * Celtiberian
 * Gallaecian
 * Nuclear Celtic?
 * Cisalpine Celtic: Lepontic → Cisalpine Gaulish
 * Core Celtic (secure)
 * Transalpine Gaulish ("Transalpine Celtic")
 * Insular Celtic
 * Goidelic
 * Brittonic

Eska considers a division of Transalpine–Goidelic–Brittonic into Transalpine and Insular Celtic to be most probable because of the greater number of innovations in Insular Celtic than in P-Celtic, and because the Insular Celtic languages were probably not in great enough contact for those innovations to spread as part of a sprachbund. However, if they have another explanation (such as an SOV substratum language), then it is possible that P-Celtic is a valid clade, and the top branching would be:

 * Core Celtic (P-Celtic hypothesis)
 * Goidelic
 * Gallo-Brittonic
 * Transalpine Gaulish ("Transalpine Celtic")
 * Brittonic

Italo-Celtic
Within the Indo-European family, the Celtic languages have sometimes been placed with the Italic languages in a common Italo-Celtic subfamily. This hypothesis fell somewhat out of favour after reexamination by American linguist Calvert Watkins in 1966. Irrespectively, some scholars such as Ringe, Warnow and Taylor and many others have argued in favour of an Italo-Celtic grouping in 21st century theses.

Characteristics
Although there are many differences between the individual Celtic languages, they do show many family resemblances.

 * consonant mutations (Insular Celtic only)
 * inflected prepositions (Insular Celtic only)
 * two grammatical genders (modern Insular Celtic only; Old Irish and the Continental languages had three genders, although Gaulish may have merged the neuter and masculine in its later forms)
 * a vigesimal number system (counting by twenties)
 * Cornish hwetek ha dew ugens "fifty-six" (literally "sixteen and two twenty")
 * verb–subject–object (VSO) word order (probably Insular Celtic only)
 * an interplay between the subjunctive, future, imperfect, and habitual, to the point that some tenses and moods have ousted others
 * an impersonal or autonomous verb form serving as a passive or intransitive
 * Welsh dysgaf "I teach" vs. dysgir "is taught, one teaches"
 * Irish múinim "I teach" vs. múintear "is taught, one teaches"
 * no infinitives, replaced by a quasi-nominal verb form called the verbal noun or verbnoun
 * frequent use of vowel mutation as a morphological device, e.g. formation of plurals, verbal stems, etc.
 * use of preverbal particles to signal either subordination or illocutionary force of the following clause
 * mutation-distinguished subordinators/relativisers
 * particles for negation, interrogation, and occasionally for affirmative declarations
 * pronouns positioned between particles and verbs
 * lack of simple verb for the imperfective "have" process, with possession conveyed by a composite structure, usually BE + preposition
 * Cornish Yma kath dhymm "I have a cat", literally "there is a cat to me"
 * Welsh Mae cath gyda fi "I have a cat", literally "a cat is with me"
 * Irish Tá cat agam "I have a cat", literally "there is a cat at me"
 * use of periphrastic constructions to express verbal tense, voice, or aspectual distinctions
 * distinction by function of the two versions of BE verbs traditionally labelled substantive (or existential) and copula
 * bifurcated demonstrative structure
 * suffixed pronominal supplements, called confirming or supplementary pronouns
 * use of singulars or special forms of counted nouns, and use of a singulative suffix to make singular forms from plurals, where older singulars have disappeared

Examples:
 * Ná bac le mac an bhacaigh is ní bhacfaidh mac an bhacaigh leat.
 * (Literal translation) Do not bother with son the beggar's and not will-bother son the beggar's with-you.
 * bhacaigh is the genitive of bacach. The igh the result of affection; the is the lenited form of.
 * leat is the second person singular inflected form of the preposition le.
 * The order is verb–subject–object (VSO) in the second half. Compare this to English or French (and possibly Continental Celtic) which are normally subject–verb–object in word order.

 * pedwar ar bymtheg a phedwar ugain
 * (Literally) four on fifteen and four twenties
 * bymtheg is a mutated form of pymtheg, which is pump ("five") plus deg ("ten"). Likewise, phedwar is a mutated form of pedwar.
 * The multiples of ten are deg, ugain, deg ar hugain, deugain, hanner cant, trigain, deg a thrigain, pedwar ugain, deg a phedwar ugain, cant.

Comparison table
The lexical similarity between the different Celtic languages is apparent in their core vocabulary, especially in terms of actual pronunciation. Moreover, the phonetic differences between languages are often the product of regular sound change (i.e. lenition of into  or Ø).

The table below has words in the modern languages that were inherited direct from Proto-Celtic, as well as a few old borrowings from Latin that made their way into all the daughter languages. There is often a closer match between Welsh, Breton and Cornish on the one hand and Irish, Scottish Gaelic and Manx on the other. For a fuller list of comparisons, see the Swadesh list for Celtic. † Borrowings from Latin.

Examples
Article 1 of the Universal Declaration of Human Rights: ''All human beings are born free and equal in dignity and rights. They are endowed with reason and conscience and should act towards one another in a spirit of brotherhood.''
 * Saolaítear gach duine den chine daonna saor agus comhionann i ndínit agus i gcearta. Tá bua an réasúin agus an choinsiasa acu agus ba cheart dóibh gníomhú i dtreo a chéile i spiorad an bhráithreachais.
 * Ta dagh ooilley pheiagh ruggit seyr as corrym ayns ard-cheim as kiartyn. Ren Jee feoiltaghey resoon as cooinsheanse orroo as by chair daue ymmyrkey ry cheilley myr braaraghyn.
 * Tha gach uile dhuine air a bhreith saor agus co-ionnan ann an urram 's ann an còirichean. Tha iad air am breith le reusan is le cogais agus mar sin bu chòir dhaibh a bhith beò nam measg fhèin ann an spiorad bràthaireil.
 * Dieub ha par en o dellezegezh hag o gwirioù eo ganet an holl dud. Poell ha skiant zo dezho ha dleout a reont bevañ an eil gant egile en ur spered a genvreudeuriezh.
 * Genys frank ha par yw oll tus an bys yn aga dynita hag yn aga gwiryow. Enduys yns gans reson ha kowses hag y tal dhedha omdhon an eyl orth y gila yn spyrys a vrederedh.
 * Genir pawb yn rhydd ac yn gydradd â'i gilydd mewn urddas a hawliau. Fe'u cynysgaeddir â rheswm a chydwybod, a dylai pawb ymddwyn y naill at y llall mewn ysbryd cymodlon.

Possible members of the family
Several poorly-documented languages may have been Celtic.

 * Ancient Belgian
 * Camunic is an extinct language spoken in the first millennium BC in the Val Camonica and Valtellina valleys of the Central Alps. It has recently been proposed that it was a Celtic language.
 * Ivernic
 * Ligurian, on the Northern Mediterranean Coast straddling the southeast French and northwest Italian coasts, including parts of Tuscany, Elba and Corsica. Xavier Delamarre argues that Ligurian was a Celtic language similar to Gaulish. The Ligurian-Celtic question is also discussed by Barruol (1999). Ancient Ligurian is listed as either Celtic (epigraphic), or Para-Celtic (onomastic).
 * Lusitanian, spoken in the area between the Douro and Tagus rivers of western Iberia (a region straddling the present border of Portugal and Spain). Known from only five inscriptions and various place names. It is an Indo-European language and some scholars have proposed that it may be a para-Celtic language that evolved alongside Celtic or formed a dialect continuum or sprachbund with Tartessian and Gallaecian. This is tied to a theory of an Iberian origin for the Celtic languages. It is also possible that the Q-Celtic languages alone, including Goidelic, originated in western Iberia (a theory that was first put forward by Edward Lhuyd in 1707) or shared a common linguistic ancestor with Lusitanian. Secondary evidence for this hypothesis has been found in research by biological scientists, who have identified (1) deep-rooted similarities in human DNA found precisely in both the former Lusitania and Ireland, and; (2) the so-called "Lusitanian distribution" of animals and plants unique to western Iberia and Ireland. Both phenomena are now generally thought to have resulted from human emigration from Iberia to Ireland, in the late Paleolithic or early Mesolithic eras. Other scholars see greater linguistic affinities between Lusitanian, Old Gallo-Italic (particularly with Ligurian) and Old European. Prominent modern linguists such as Ellis Evans, believe Gallaecian-Lusitanian was in fact one same language (not separate languages) of the "P" Celtic variant.
 * Rhaetic, spoken in central Switzerland, Tyrol in Austria, and the Alpine regions of northeast Italy. Documented by a limited number of short inscriptions (found through Northern Italy and Western Austria) in two variants of the Etruscan alphabet. Its linguistic categorization is not clearly established, and it presents a confusing mixture of what appear to be Etruscan, Indo-European, and uncertain other elements. Howard Hayes Scullard argues that Rhaetian was also a Celtic language.
 * Tartessian, spoken in the southwest of the Iberia Peninsula (mainly southern Portugal and southwest Spain). Tartessian is known by 95 inscriptions, with the longest having 82 readable signs.  John T. Koch argues that Tartessian was also a Celtic language.

title: Celtic languages