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Nephroid

In geometry, a nephroid is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger one by a factor of one-half.

Name
Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Richard A. Proctor in 1878.

Strict definition
A nephroid is
 * an algebraic curve of degree 6.
 * an epicycloid with two cusps
 * a plane simple closed curve = a Jordan curve

Parametric
If the small circle has radius $$a$$, the fixed circle has midpoint $$(0,0)$$ and radius $$2a$$, the rolling angle of the small circle is $$2\varphi$$ and point $$(2a,0)$$ the starting point (see diagram) then one gets the parametric representation:
 * $$x(\varphi) = 3a\cos\varphi- a\cos3\varphi=6a\cos\varphi-4a \cos^3\varphi \ ,$$
 * $$y(\varphi) = 3a \sin\varphi - a\sin3\varphi =4a\sin^3\varphi\, \qquad 0\le \varphi < 2\pi$$

The complex map $$z \to z^3 + 3z$$ maps the unit circle to a nephroid

Proof of the parametric representation
The proof of the parametric representation is easily done by using complex numbers and their representation as complex plane. The movement of the small circle can be split into two rotations. In the complex plane a rotation of a point $$z$$ around point $$0$$ (origin) by an angle $$\varphi$$ can be performed by the multiplication of point $$z$$ (complex number) by $$ e^{i\varphi}$$. Hence the
 * rotation $$\Phi_3$$ around point $$3a$$ by angle $$2\varphi$$ is $$: z \mapsto 3a+(z-3a)e^{i2\varphi}$$ ,
 * rotation $$\Phi_0$$ around point $$0$$ by angle $$\varphi$$ is $$:\quad z \mapsto ze^{i\varphi}$$.

A point $$ p(\varphi)$$ of the nephroid is generated by the rotation of point $$2a$$ by  $$\Phi_3$$ and the subsequent rotation with $$\Phi_0$$:
 * $$p(\varphi)=\Phi_0(\Phi_3(2a))=\Phi_0(3a-ae^{i2\varphi})=(3a-ae^{i2\varphi})e^{i\varphi}=3ae^{i\varphi}-ae^{i3\varphi}$$.

Herefrom one gets

\begin{array}{cclcccc} x(\varphi)&=&3a\cos\varphi-a\cos3\varphi &=& 6a\cos\varphi-4a \cos^3\varphi \ ,&& \\ y(\varphi)&=&3a\sin\varphi-a\sin3\varphi&=& 4a\sin^3\varphi &.& \end{array} $$ (The formulae $$ e^{i\varphi}=\cos\varphi+ i\sin\varphi, \ \cos^2\varphi+ \sin^2\varphi=1, \ \cos3\varphi=4\cos^3\varphi-3\cos\varphi,\;\sin 3\varphi=3\sin\varphi -4\sin^3\varphi$$ were used. See trigonometric functions.)

Implicit
Inserting $$x(\varphi)$$ and $$y(\varphi)$$ into the equation shows that this equation is an implicit representation of the curve.
 * $$(x^2+y^2-4a^2)^3=108a^4y^2$$

Proof of the implicit representation
With
 * $$x^2+y^2-4a^2=(3a\cos\varphi-a\cos3\varphi)^2+(3a\sin\varphi-a\sin3\varphi)^2 -4a^2=\cdots=6a^2(1-\cos2\varphi)=12a^2\sin^2\varphi$$

one gets
 * $$(x^2+y^2-4a^2)^3=(12a^2)^3\sin^6\varphi=108a^4(4a\sin^3\varphi)^2=108a^4y^2\ .$$

Orientation
If the cusps are on the y-axis the parametric representation is
 * $$x=3a\cos \varphi+a\cos3\varphi,\quad y=3a\sin \varphi+a\sin3\varphi).$$

and the implicit one:
 * $$(x^2+y^2-4a^2)^3=108a^4x^2.$$

Metric properties
For the nephroid above the The proofs of these statements use suitable formulae on curves (arc length, area and radius of curvature) and the parametric representation above
 * arclength is $$ L= 24 a, $$
 * area $$ A= 12\pi a^2\ $$ and
 * radius of curvature is $$\rho=|3a\sin \varphi|.$$
 * $$x(\varphi)=6a\cos\varphi-4a \cos^3\varphi \, $$
 * $$y(\varphi)= 4a\sin^3\varphi $$

and their derivatives
 * $$\dot x=-6a\sin\varphi(1 - 2\cos^2\varphi)\ ,\quad \ \ddot x= -6 a\cos \varphi(5-6\cos^2\varphi)\ ,$$
 * $$\dot y=12a\sin^2\varphi\cos\varphi \quad, \quad \quad \quad \quad

\ddot y=12a\sin\varphi(3\cos^2\varphi-1)\. $$

 * Proof for the arc length:
 * $$L=2\int_0^\pi{\sqrt{\dot x^2+\dot y^2}} \; d\varphi=\cdots =12a\int_0^\pi \sin\varphi\; d\varphi= 24a$$.

 * Proof for the area:
 * $$ A=2\cdot \tfrac{1}{2}|\int_0^\pi[x \dot y-y \dot x]\; d\varphi|=\cdots= 24a^2\int_0^\pi\sin^2\varphi\; d\varphi= 12\pi a^2$$.

 * Proof for the radius of curvature:
 * $$\rho = \left|\frac {\left({\dot{x}^2 + \dot{y}^2}\right)^\frac32}{\dot {x}\ddot{y} - \dot{y}\ddot{x}}\right|=\cdots= |3a\sin \varphi|.$$

Construction

 * It can be generated by rolling a circle with radius $$a$$ on the outside of a fixed circle with radius $$2a$$. Hence, a nephroid is an epicycloid.

Nephroid as envelope of a pencil of circles

 * Let be $$c_0$$ a circle and $$D_1,D_2$$ points of a diameter $$d_{12}$$, then the envelope of the pencil of circles, which have midpoints on $$c_0$$ and are touching $$d_{12}$$ is a nephroid with cusps $$D_1,D_2$$.

Proof
Let $$c_0$$ be the circle $$(2a\cos\varphi,2a\sin\varphi)$$ with midpoint $$(0,0)$$ and radius $$2a$$. The diameter may lie on the x-axis (see diagram). The pencil of circles has equations:
 * $$ f(x,y,\varphi)=(x-2a\cos\varphi)^2+(y-2a\sin\varphi)^2-(2a\sin\varphi)^2=0 \ .$$

The envelope condition is
 * $$f_\varphi(x,y,\varphi)=2a(x\sin\varphi -y\cos\varphi-2a\cos\varphi\sin\varphi)=0\ . $$

One can easily check that the point of the nephroid $$p(\varphi)=(6a\cos\varphi-4a \cos^3\varphi\; ,\; 4a\sin^3\varphi)$$ is a solution of the system $$f(x,y,\varphi)=0, \; f_\varphi(x,y,\varphi)=0$$ and hence a point of the envelope of the pencil of circles.

Nephroid as envelope of a pencil of lines
Similar to the generation of a cardioid as envelope of a pencil of lines the following procedure holds:
 * 1) Draw a circle, divide its perimeter into equal spaced parts with $$3N$$ points (see diagram) and number them consecutively.
 * 2) Draw the chords: $$(1,3), (2,6), ...., (n,3n),...., (N,3N), (N+1,3), (N+2,6), ...., $$. (i.e.: The second point is moved by threefold velocity.)
 * 3) The envelope of these chords is a nephroid.

Proof
The following consideration uses trigonometric formulae for $$ \cos \alpha+\cos\beta,\ \sin \alpha+\sin\beta, \ \cos (\alpha+\beta), \ \cos2\alpha$$. In order to keep the calculations simple, the proof is given for the nephroid with cusps on the y-axis. Equation of the tangent: for the nephroid with parametric representation
 * $$x=3\cos\varphi + \cos3\varphi,\; y=3\sin\varphi+\sin3\varphi$$:

Herefrom one determines the normal vector $$\vec n=(\dot y, -\dot x)^T $$, at first.

The equation of the tangent $$\dot y(\varphi)\cdot (x -x(\varphi)) - \dot x(\varphi)\cdot (y-y(\varphi))= 0$$ is:
 * $$(\cos2\varphi\cdot x \ + \ \sin 2\varphi\cdot y)\cos \varphi = 4\cos^2 \varphi \ .$$

For $$ \varphi=\tfrac{\pi}{2},\tfrac{3\pi}{2}$$ one gets the cusps of the nephroid, where there is no tangent. For $$ \varphi\ne\tfrac{\pi}{2},\tfrac{3\pi}{2}$$ one can divide by $$\cos\varphi$$ to obtain
 * $$\cos2\varphi \cdot x +  \sin2\varphi \cdot y = 4 \cos\varphi \ .$$

Equation of the chord: to the circle with midpoint $$(0,0)$$ and radius $$4$$: The equation of the chord containing the two points $$(4\cos\theta, 4\sin\theta), \ (4\cos{\color{red}3}\theta, 4\sin{\color{red}3}\theta)) $$ is:
 * $$(\cos2\theta \cdot x +  \sin2\theta \cdot y)\sin\theta = 4 \cos\theta\sin\theta \ .$$

For $$\theta =0, \pi$$ the chord degenerates to a point. For $$\theta \ne 0,\pi$$ one can divide by $$\sin\theta$$ and gets the equation of the chord: The two angles $$\varphi, \theta$$ are defined differently ($$\varphi$$ is one half of the rolling angle, $$\theta$$ is the parameter of the circle, whose chords are determined), for $$\varphi=\theta $$ one gets the same line. Hence any chord from the circle above is tangent to the nephroid and
 * $$\cos2\theta \cdot x +  \sin2\theta \cdot y = 4 \cos\theta \ .$$
 * the nephroid is the envelope of the chords of the circle.

Nephroid as caustic of one half of a circle

The considerations made in the previous section give a proof for the fact, that the caustic of one half of a circle is a nephroid.

 * If in the plane parallel light rays meet a reflecting half of a circle (see diagram), then the reflected rays are tangent to a nephroid.

Proof
The circle may have the origin as midpoint (as in the previous section) and its radius is $$4$$. The circle has the parametric representation
 * $$k(\varphi)=4(\cos\varphi,\sin\varphi) \ .$$

The tangent at the circle point $$K:\ k(\varphi)$$ has normal vector  $$\vec n_t=(\cos\varphi,\sin\varphi)^T$$. The reflected ray has the normal vector (see diagram) $$\vec n_r=(\cos{\color{red}2}\varphi,\sin{\color{red}2}\varphi)^T$$ and containing circle point $$K: \ 4(\cos\varphi,\sin\varphi) $$. Hence the reflected ray is part of the line with equation
 * $$\cos{\color{red}2}\varphi\cdot x \ + \ \sin {\color{red}2}\varphi\cdot y = 4\cos\varphi \ ,$$

which is tangent to the nephroid of the previous section at point
 * $$P:\ (3\cos\varphi + \cos3\varphi,3\sin\varphi+\sin3\varphi)$$ (see above).

The evolute and involute of a nephroid
[[File:Nephroide-evol.svg|300px|thumb|nephroid and its evolute

magenta: point with osculating circle and center of curvature]]

Evolute
The evolute of a curve is the locus of centers of curvature. In detail: For a curve $$\vec x=\vec c(s)$$ with radius of curvature $$\rho(s)$$ the evolute has the representation
 * $$\vec x=\vec c(s) + \rho(s)\vec n(s).$$

with $$\vec n(s)$$ the suitably oriented unit normal.

For a nephroid one gets:
 * The evolute of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram).

Proof
The nephroid as shown in the picture has the parametric representation
 * $$x=3\cos\varphi + \cos3\varphi,\quad y=3\sin\varphi+\sin3\varphi \ ,$$

the unit normal vector pointing to the center of curvature
 * $$\vec n(\varphi)=(-\cos 2\varphi,-\sin 2\varphi)^T$$ (see section above)

and the radius of curvature $$3\cos \varphi$$ (s. section on metric properties). Hence the evolute has the representation:
 * $$x=3\cos\varphi + \cos3\varphi -3\cos\varphi\cdot\cos2\varphi=\cdots=3\cos\varphi-2\cos^3\varphi,$$
 * $$y=3\sin\varphi+\sin3\varphi -3\cos\varphi\cdot\sin2\varphi\ =\cdots=2\sin^3\varphi \ ,$$

which is a nephroid half as large and rotated 90 degrees (see diagram and section  above)

Involute
Because the evolute of a nephroid is another nephroid, the involute of the nephroid is also another nephroid. The original nephroid in the image is the involute of the smaller nephroid.

Inversion of a nephroid
The inversion
 * $$ x \mapsto \frac{4a^2x}{x^2+y^2}, \quad y\mapsto \frac{4a^2y}{x^2+y^2} $$

across the circle with midpoint $$(0,0)$$ and radius $$2a$$ maps the nephroid with equation
 * $$(x^2+y^2-4a^2)^3=108a^4y^2$$

onto the curve of degree 6 with equation
 * $$(4a^2-(x^2+y^2))^3=27a^2(x^2+y^2)y^2$$ (see diagram).

title: Nephroid


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Little Torrington

Little Torrington is a village and a civil parish near Great Torrington, in the Torridge district, north Devon, England. In 2001 the population of the civil parish of Little Torrington was 420 and in 2011 it was 376, according to census data. Little Torrington has the Church of St Giles and the Chapel of St Mary Magdalene.

In the 1870s, Little Torrington was described as follows:"'A bridge over the river Torridge, at Taddyport village, connects the parish with Great Torrington. The living is a rectory in the diocese of Exeter. Value, £397. Patrons, the Heirs of Lord Rolle and others. The church is ancient but good.'"

History
Little Torrington was originally a Saxon settlement called Toritona, held by Edmer Ator during the reign of Edward the Confessor between 1042–1066.

At the time of the Norman Invasion of England in 1066, Alweard the Red was lord and tenant. By 1086 Little Torrington was considered a large settlement for its time with 33 households. It had a new overlord, Count Rotbert of Mortrain, the uterine half-brother of William the Conqueror. The local tenant-in-chief was also from the Norman elite, Aluredus le butuiller (Alfred the Butler).

The first vicar for the church was recorded in 1259, although the current church of St Giles was created in the 15th century. Renovations have taken place on the church throughout its time. The first major renovation took place in 1755, when it was re-roofed in a Georgian style. In 1857, the nave and south aisle were restored, the western gallery removed, and the chancel rebuilt and enlarged. Finally, in 1898, restoration work was carried out on the tower, and a tower screen and new pews were added to the interior of the church.

Little Torrington, along with Great Torrington, had a significant role in the English Civil War; the Battle of Torrington in 1646 marked the end of Royalist resistance in the region.

The population of Little Torrington fluctuated between 1801 and 2011 according to census data. In 1801 it was 449, and from then the population continued to grow, reaching a peak of 623 residents in 1851. The population then fell and reached its lowest recorded level of 281 in 1961. The population in 2011 had recovered to 376 people.

Industry

The first census divided the working men into two different categories, those "chiefly employed in agriculture" and those "chiefly employed in trade, manufacturers or handicraft". This increased to nine categories in 1831, with the majority of Little Torrington's men aged 20 and over working in agriculture.

The 1881 census showed more work categories for both men and women. Agriculture remained the main industry for men, and dressmaking for women; the second largest category for both genders was domestic and office services.

As of 2011, 189 of 376 Little Torrington residents were employed. The highest areas of employment were agriculture, forestry, fishing and manufacturing.

Geography
The village is located east of the A386, contains "2880 acres of land, including the village of Taddiport, on the south-west side of the river Torridge opposite Great Torrington."

"Taddiport is a hamlet beside the Torridge, which is here crossed by a massive three-arched bridge, possibly of 17th century date. A chapel, attached to a leper hospital founded in the 13th century, still remains, though considerably altered."

The majority of current residents, according to the 2011 census, live in whole houses or bungalows, 106 households live in detached properties and 55 households live in semi-detached properties, with only 2 households living in flats or apartments.

Out of all the current residents 189 are economically active and 70 are retired. The majority of these 178 households contain 2 people, 85 households, 51 households only have 1 person and only 14 households contain 4 people.

According to the 2001 census age structure indicators, they found out of the 420 people living in Little Torrington in 2001, 112 are aged 45–59, 71 are aged 30–44, 29 are aged 75–84 and only 12 are aged 5–7.

There are 342 residents aged 16 or over, 99 of these residents have no formal qualifications, 41 have GCSEs or equivalent, 45 have A-Levels or equivalent and 78 have degrees or equivalent as their highest level of qualification.

Local businesses

 * The Torridge Inn – located in Taddiport, this is a local rural pub and authentic Thai diner, that provides live music and entertainment for patrons.
 * Tarka Pottery – a pottery studio located on Glebe Farm, which has been open since 2006 offering clay workshops, classes and pottery painting for all ages. The studio is run by Jax Williams who has been working with clay for over 25 years and also runs a holiday cottage for holidays and pottery breaks.
 * Smytham Manor – a family run, rural holiday park that provides camping, caravanning and cabin accommodations. The site is set on "23 acres of parkland adjoining to the old manor house, which is still used as a family home".

Transport
Little Torrington is connected by various bus services including those going to Barnstaple and Exeter.

The closest railway station is in Chapelton, 10 miles from Little Torrington, part of the First Great Western line which connects to Barnstaple, Exeter and Plymouth.

Historic estates

 * Cross, Little Torrington

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Little Torrington


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Titan IIIC

The Titan IIIC was an expendable launch system used by the United States Air Force from 1965 until 1982. It was the first Titan booster to feature large solid rocket motors and was planned to be used as a launcher for the Dyna-Soar, though the spaceplane was cancelled before it could fly. The majority of the launcher's payloads were DoD satellites, for military communications and early warning, though one flight (ATS-6) was performed by NASA. The Titan IIIC was launched exclusively from Cape Canaveral while its sibling, the Titan IIID, was launched only from Vandenberg AFB.

History
The Titan rocket family was established in October 1955 when the Air Force awarded the Glenn L. Martin Company (later Martin Marietta and now Lockheed Martin) a contract to build an intercontinental ballistic missile (SM-68). It became known as the Titan I, the nation's first two-stage ICBM, and replaced the Atlas ICBM as the second underground, vertically stored, silo-based ICBM. Both stages of the Titan I used kerosene (RP-1) and liquid oxygen (LOX) as propellants. A subsequent version of the Titan family, the Titan II, was similar to the Titan I, but was much more powerful. Designated as LGM-25C, the Titan II was the largest USAF missile at the time and burned Aerozine 50 and nitrogen tetroxide (NTO) rather than RP-1 and LOX.

The Titan III family consisted of an enhanced Titan II core with or without solid rocket strap-on boosters and an assortment of upper stages. All Solid Rocket Motor (SRM)-equipped Titans (IIIC, IIID, IIIE, 34D, and IV) launched with only the SRMs firing at liftoff, the core stage not activating until T+105 seconds, shortly before SRM jettison. The Titan IIIA (an early test variant flown in 1964–65) and IIIB (flown from 1966 to 1987 with an Agena D upper stage in both standard and extended tank variants) had no SRMs. The Titan III launchers provided assured capability and flexibility for launch of large-class payloads.

All Titan II/III/IV vehicles contained a special range safety system known as the Inadvertent Separation Destruction System (ISDS) that would activate and destroy the first stage if there was a premature second stage separation. Titans that carried Solid Rocket Boosters (SRBs) (Titan IIIC, IIID, 34D, and IV) had a second ISDS that consisted of several lanyards attached to the SRBs that would trigger and automatically destroy them if they prematurely separated from the core, said "destruction" consisting mainly of splitting the casings open to release the pressure inside and terminate thrust. The ISDS would end up being used a few times over the Titan's career.

Another slight modification to SRB-equipped Titans was the first stage engines being covered instead of the open truss structure on the Titan II/IIIA/IIIB. This was to protect the engines from the heat of the SRB exhaust.

Titan III/IV SRBs were fixed nozzle and for roll control, a small tank of nitrogen tetroxide was mounted to each motor. The would be injected into the SRB exhaust to deflect it in the desired direction.

As the IIIC consisted of mostly proven hardware, launch problems were generally only caused by the upper stages and/or payload. The second launch in October 1965 failed when the Transtage suffered an oxidizer leak and was unable to put its payload (several small satellites) into the correct orbit. The third launch in December experienced a similar failure.

The fourth IIIC launch was used to send the LES 4 (Lincoln Experimental Satellite 4) into orbit. It was a US Air Force experimental communications satellite launched along with OV2-3, LES 3, and Oscar 4 from Cape Canaveral aboard a single Titan 3C rocket. It transmitted in X-band.

The fifth Titan IIIC (August 26, 1966) failed shortly after launch when pieces of the payload fairing started breaking off. Around 80 seconds, the remainder of the shroud disintegrated, causing loss of launch vehicle control as well as the payload (a group of IDCSP satellites intended to provide radio communication for the US Army in Vietnam). The ISDS activated automatically when one of the SRBs broke away from the stack and destroyed the entire launch vehicle. The exact reason for the shroud failure was not determined, but the fiberglass payload shrouds used on the Titan III up to this point were replaced with a metal shroud afterwards.

A Titan IIIC in November 1970 failed to place its missile early warning satellite in the correct orbit due to a Transtage failure and a 1975 launch of two DSCS II military communication satellites left in LEO by another Transtage failure.

On March 25, 1978, a launch of two DSCS II satellites ended up in the Atlantic Ocean when the Titan second stage hydraulic pump failed, resulting in engine shutdown approximately 470 seconds after launch. The Range Safety destruct command was sent, but it was unclear if the stage received it or if it had already broken up by that point.

The first Titan IIIC flew on June 18, 1965, and was the most powerful launcher used by the Air Force until it was replaced by the Titan 34D in 1982. The last IIIC was launched in March 1982.

Design

The Titan IIIC weighed about 626000 kg at liftoff and consisted of a two-stage Titan core and upper stage called the Titan Transtage, both burning hypergolic liquid fuel, and two large UA1205 solid rocket motors.

The solid motors were ignited on the ground and were designated "stage 0". Each motor composed of five segments and was 10 ft in diameter, 85 ft long, and weighed nearly 500000 lb. They produced a combined 2380000 lbf thrust at sea level and burned for approximately 115 seconds. Solid motor jettison occurred at approximately 116 seconds.

The first core stage ignited about 5 seconds before SRM jettison. Designated the Titan 3A-1, this stage was powered by a twin nozzle Aerojet LR-87-AJ9 engine that burned about 110000 kg of Aerozine 50 and nitrogen tetroxide (NTO) and produced 1941.7 kN thrust over 147 seconds. The Aerozine 50 and NTO were stored in structurally independent tanks to minimize the hazard of the two mixing if a leak should have developed in either tank.

The second core stage, the Titan 3A-2, contained about 25000 kg of propellant and was powered by a single Aerojet LR-91-AJ9, which produced 453.7 kN for 145 seconds.

The upper stage, the Titan Transtage, also burned Aerozine 50 and NTO. Its two Aerojet AJ-10-138 engines were restartable, allowing flexible orbital operations including orbital trimming, geostationary transfer and insertion, and delivery of multiple payloads to different orbits. This required complex guidance and instrumentation. Transtage contained about 10000 kg of propellant and its engines delivered 16000 lbf.

General characteristics

 * Primary Function: Space booster
 * Builder: Martin Marietta
 * Power Plant:
 * Stage 0 consists of two solid rocket motors.
 * Stage 1 uses two LR87 liquid propellant engines.
 * Stage 2 uses one LR91 liquid propellant engine.
 * Stage 3 uses two Aerojet AJ-10-138 liquid propellant engines.
 * Length: 42 m
 * Stage 0: 25.91 m
 * Stage 1: 22.28 m
 * Stage 2: 7.9 m
 * Stage 3: 4.57 m
 * Diameter:
 * Stage 0: 3.05 m
 * Stage 1: 3.05 m
 * Stage 2: 3.05 m
 * Stage 3: 3.05 m
 * Mass:
 * Stage 0: Empty 33,798 kg/ea; Full 226,233 kg/ea
 * Stage 1: Empty 5,443 kg; Full 116,573 kg
 * Stage 2: Empty 2,653 kg; Full 29,188 kg
 * Stage 3: Empty 1,950 kg; Full 12,247 kg
 * Lift capability:
 * Up to 28,900 lb (13,100 kg) into a low Earth orbit with 28 degrees inclination.
 * Up to 6,600 lb (3,000 kg) into a geosynchronous transfer orbit when launched from Cape Canaveral Air Force Station, FL.
 * Maximum takeoff weight: 626,190 kg
 * Cost:
 * Date deployed: June 1965.
 * Launch sites: Cape Canaveral Air Force Station, FL., and Vandenberg Air Force Base, CA.

title: Titan IIIC


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Black Torrington

Black Torrington is a village and civil parish in Torridge, Devon, England, situated between the towns of Holsworthy and Hatherleigh. It is located on and named after (the dark waters of) the River Torridge. In the 2021 UK census, the population of Black Torrington was recorded as having been 528.

In the Domesday book of 1066 it had a recorded population of 67 households, which put it in the largest 20% of English settlements at the time.

Buildings in the village include the grade II* listed Church of St Mary, a 15th-century Church of England parish church, and the grade II listed Coham, an 18th century country house.

title: Black Torrington


text:
Talk:List of Babylon 5 episodes

Double Asterisk Missing explanation
Could someone please add a note somewhere explaining what the double asterisk (**) for 4 episodes in season 2 means? The double asterisk in the episode/movie chronology section near the top is explained, but does not apply in the section devoted to season 2. Is this referring to the "improved episode" as referenced below? — Preceding unsigned comment added by 208.185.102.168 (talk) 18:09, 14 March 2018 (UTC)

Production Code
Can someone clarify what this is? —Preceding unsigned comment added by 146.145.251.34 (talk) 19:03, 14 October 2010 (UTC)

Somebody on purpose mixed up the production code, mostly interchanging the production code of one number with 1 less, probably so that if you download a pirated copy of Babylon 5 episodes according to this guide, you get the wrong one. I've rectified it for Season 1, 2 and 5, Season 3 & 4 seem to be unaffected, but the biggest mess was in Season 1 and Season 5. Please lock this article, so TimeWarner employees cannot change it again. — Preceding unsigned comment added by 88.84.21.77 (talk • contribs)
 * Have reverted this, you seem to be unaware of what production codes are, they are not the episode order, these episodes were produced/broadcasted out of order.  X  eworlebi (talk) 08:04, 19 June 2011 (UTC)

Remove wikipedia articles for each episode?
Maybe I'm begging for a beating by asking this here, but: as a user of wikipedia, I think the "One article per each episode" is sort of overwhelming. I don't like it for a few reasons.:
 * It clutters up the namespace (example: while searching for "The Summoning", the video game, I end up at "The Summoning", the Bab5 episode.  Given the huge number of episodes, that's significant.
 * Each individual article doesn't contain all that much data, but has lots of metadata.
 * The entire project is completely duplicative of The Lurker's Guide. If this was on a Bab5 wiki it would make sense, but I think it's a hell of a lot of overhead to jam into a general-purpose encyclopedia.

Am I entirely alone in this, or are there other people who think that this might be overdoing it a bit? If there's consensus on this, I'd like to propose the following:
 * 1) Change all wikilinks in List of Babylon 5 episodes to point to their corresponding Lurker's Guide entries.
 * 2) Mass delete all of the wikilinked per-episode articles, perhaps making an exception for the pilot movie.

Thoughts? I also cross-posted this proposal over on Talk:Babylon 5. Nandesuka 19:08, 17 July 2005 (UTC)

 * Sorry for the late reply, I only saw this today... I'd have to say that I totally disagree with changing wikilinks to point off Wikipedia because since there is no original research on Wikipedia everything here is already out there somewhere. The point of Wikipedia is collate useful information here rather than simply point people elsewhere. That said, I also don't think it's particular useful to have one article for each episode. What I think would be best is to organise the episodes into omnibus articles (as per WP:FICT) for each season, rather like they have for Lost season 1 episode guide then put the Lurkers guide in as an external link at the end. -- Lochaber 13:46, 10 August 2005 (UTC)

 * My thoughts: Many (or most) major TV series have an article for each episode. It seems a bit inconsistent to delete the entire B5 articles, especially as B5 is notable as a landmark series.  Some of the episodes are highly notable, some having beaten major film productions to win Hugo Awards.
 * Additionally, as B5 made a major contribution, not just to science fiction, but to the entire genre of television drama as a whole, as a major pioneer of arc-driven overarching storylines, it could very well be argued that having individual articles is a good way (or even a necessary way) to show this.
 * Rather than delete, I think we should work on slowly recreating or restoring them (archive.org) as Stub articles and mark them with improvement templates, and deleting any un-encyclopedic content. The aim of Wikipedia is to be a repository of human knowledge, so I feel we should aim for improvement, rather than deletion.  Stub articles are okay on Wikipedia.  I suggest that these be restored, with real-world content being added, with good sources being DVD commentaries, interviews, and the "jms speaks" sections of http://www.midwinter.com/lurk/ etc, as these are Straczynski speaking himself.  While DVD commentaries are not available for all episodes, some of the comments are applicable to multiple episodes.  Fan sites by individuals (especially those adding rather fictional information) aren't usually helpful.
 * I know the conflict of page names issue is annoying. If I understand correctly this ought to be fixed via renaming e.g. "Chrysalis (Babylon 5 episode)", especially for titles which are commonly used words or phrases (e.g. chrysalis, knives, endgame etc); or via use of disambiguation pages, rather than by deletion.
 * Whilst a Wikipedia plot summary ought not simply duplicate one published elsewhere, it would be expected that Wikipedia would duplicate much information published in the Lurker's guide, such as J. Michael Straczynski's comments in the "jms speaks" sections, as these are aggregations of his statements from several different places. As it was made in the 90s, there wasn't as much published real-world information as recent TV series, so we can only do what we can. — Preceding unsigned comment added by Fh1 (talk • contribs) 23:00, 28 November 2020 (UTC)

--2003:C1:C705:9100:246C:19A1:3BD8:AA1B (talk) 13:30, 25 July 2022 (UTC)
 * hello, first i find it sad that the episodes pages have been removed, but links still exist and return one to the listings page. second, if b5 loses it's per-episode-pages then this needs to happen to other series as well! start with Game of Thrones!

Updated Episode List Style
I've updated the first season episode list to a style in fitting with other episode lists on wikipedia. Unless anyone has any serious objections, I'll continue with it when I have some more time.--Warpfactor 11:47, 17 August 2007 (UTC)

 * I've done up the season 2 list in the same style. Additionally, I added different coloured lines to each season, based on the DVD colour schemes. But if you want to redo the episode descriptions, I'll understand. -- Bronzethumb (talk) 11:56, 20 January 2008 (UTC)

Mentioning the improved episode order?
Shouldn't it be mentioned that there exists another order which actually improves it? The Lurker's Guide assembled it based on JMS's posts on the Internet. 81.182.236.91 (talk) 15:53, 13 May 2009 (UTC)

I'm confused as it is, that when the article tells me that some episodes in Season 2 have been broadcast in the wrong order (even JMS confirms it) that the episode order isn't fixed. --IvoPalli (talk) 08:43, 5 February 2023 (UTC)

 * I'm not sure what you mean. The table lists the episodes in the order in which they aired, while the note in the lead mentions that the original airing sequence for a couple of episodes isn't the intended chronological sequence. Where's the confusion? DonIago (talk) 16:49, 6 February 2023 (UTC)

 * What is the practical use of have a list of episodes in the aired sequence instead of having them in the sequence in the way they were meant to be viewed? --2A02:A443:875D:1:9DF9:472:2B18:31FC (talk) 11:34, 9 February 2023 (UTC)
 * Well, that goes back to the fundamental question of whether one believes it's more important to present the episodes in the order in which they were actually aired, the order in which they were originally produced, or the order in which the showrunner intended for them to be seen. This is a perennial problem with Futurama episodes, as editors frequently edit them to list the episodes' airing order rather than production order.
 * I would suggest that the airing order be retained but the bit about them being out of sequence with JMS's recommended order be included as a footnote rather than in a bit of prose that's completely separate from the episode listing. DonIago (talk) 14:01, 9 February 2023 (UTC)

Season 5 episode 1
Why is it that there doesn't seem to be one?
 * The short answer is simple: There is.  It is merely called 502 rather than 501.  The long answer is hard to explain:  The episode now known as 523 was originally produced as 422.  However, when the fifth season was greenlighted, a "new" 422 had to be written and produced.  This left one too many episodes.  Apparently the original 422 got pushed to 501, couldn't stay there, but once the first episode of the fifth season had to be numbered for production reasons, became 502.  The old 501 became 523, as it *has* to go last (and JMS admits in DVD commentary he did not finish post-production on 523 for emotional reasons until the last possible moment prior to air).  (Man, somebody clean up my explanation so it makes sense.)  (Also, the episode shuffling is explained in note 1 in the article.)  --Aladdin Sane (talk) 21:22, 21 September 2009 (UTC)

Original air date incorrect for late S1-3
Here in the UK, with the exception of S5, each season of B5 ran straight through at one episode per week, so, for each of the first three seasons, the season finale (and some preceding episodes) aired here in the UK before the date listed as the original air date. S4 started late enough in the year to finish after the US, and S5 had a 5-month gap between episodes 17 and 18 over here to avoid overtaking.

Should there be some indication on the page that it's factually inaccurate?

Rmsgrey (talk) 12:34, 11 July 2014 (UTC)

External links modified
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Merge
The episode "In the Kingdom of the Blind" is already among those listed here. There is not that much info in the "In the Kingdom of the Blind" article and it could either be merged or a redirect. Mannanan51 (talk) 03:42, 30 April 2017 (UTC)
 * Support unless that article can be bulked up with real-world information other than mere reviews. DonIago (talk) 04:05, 30 April 2017 (UTC)
 * ✅, along with "Phoenix Rising (Babylon 5)" which had even less content. – Fayenatic  L ondon 21:21, 5 May 2017 (UTC)

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Season 4 Episode 1 "The Hour of the Wolf"
The link to this episode's page redirects back to the link despite the page briefly appearing on-screen. Could someone fix this please? Thanks in advance Gowt (talk) 04:58, 22 February 2018 (UTC)
 * I'm not sure what you'd like changed. The episode article was redirected after a move discussion and because the page for the article contained no substantive content. If you'd like there to be an article for the episode, you can go to the article and add substantive content in place of the redirect (though I'd recommend having others review it before doing so). Alternately, you could unlike the article from the list, since as you noted it's currently a circular link. DonIago (talk) 15:17, 22 February 2018 (UTC)

Films
I see that Babylon 5: The Gathering, Babylon 5: The Legend of the Rangers, and Babylon 5: The Lost Tales are on this page. Shouldn't Babylon 5: In the Beginning, Babylon 5: Thirdspace, and Babylon 5: The River of Souls also be added? Morriswa (Charlotte Allison) (talk) 09:02, 10 March 2019 (UTC)
 * Thirdspace does appear, but it's listed under season five. Not sure what's going on with this listing, honestly. I think all of the films should be listed, but probably outside the flow of the episodes (in other words, we shouldn't be listing things "chronologically"). DonIago (talk) 13:31, 10 March 2019 (UTC)
 * All the movies are listed on this page. They were in a separate section until last May, when they were moved into the episodes section. - Eureka Lott 15:09, 11 March 2019 (UTC)
 * thank you for your responses. Even though the films may be connected to certain seasons, they should not be included in the season sections. They should all, except for maybe The Gathering (since it is the pilot film), be moved into a separate film section. Morriswa (Charlotte Allison) (talk) 16:21, 11 March 2019 (UTC)
 * as the editor who merged the films into the episode list, would you like to weigh in on this? My feeling is that this list should reflect an out-of-universe perspective (though perhaps with appropriate and sourced notes as to where the films may fit in-universe). In any event, merging the films with the standard episodes in airdate order seems very unintuitive to me. DonIago (talk) 05:00, 12 March 2019 (UTC)
 * I agree with your comment/question to User:TVBuff90. As I have noticed on other franchise pages, the films are not usually included with the season sections, unless very closely connected. Morriswa (Charlotte Allison) (talk) 14:00, 12 March 2019 (UTC)
 * Given that they're not responding but also aren't inactive, you have my blessing to break the films out. DonIago (talk) 20:44, 18 March 2019 (UTC)

TV movies
Applied the changes to the way it was suggested in the above discussion, but never done for some reason. Note, this isn't like Battlestar Galactica: Razor, where the Movie was produced as part of the season. These were separate productions, even though they were concurrently released with Season 5. I'd keep The Gathering separate, because it was the pilot and not a spin-off like the others, even though it was technically also a TV-movie (the heading can be changed, but I didn't have a good idea). furthermore I don't now the production code for The Legend of Rangers and think it should either be added or the codes should be completely left out. Any help would be great. Have a nice day, stay healthy.2003:E0:6F25:1249:99C8:A9AB:F6E:2CE1 (talk) 06:57, 30 January 2021 (UTC)

The Road Home is not a TV movie
I think The Road Home shouldn't be listed as a TV movie. While technically it's not a "DVD", I think it's still closer to The Lost Tales in categorization than to the TNT TV specials and TV pilots. Suggestion: change the title "Babylon 5: The Lost Tales" to "Direct-to-video films" and move The Road Home from Television films list to the Direct-to-video films list. Also, I think the two Voices in the Dark stories should be merged into one row. 80.98.191.91 (talk) 13:18, 8 January 2024 (UTC)

 * Another option would be to rename "Television films" to simply "Films", and perhaps update the descriptions of the films that were made for television accordingly. As it is, I agree that it's misleading to have Road Home under "Television films" as it wasn't made for television.
 * I don't think I have a strong opinion at this time as to whether the two "Voices in the Dark" stories should be consolidated. DonIago (talk) 13:47, 8 January 2024 (UTC)
 * Made the boldest version of merging things for everyone to see how it looks and see how much of it should stay. Sorry it's ugly, I'm not very good at this. I also added the episode titles to the pilot movies since this article is an episode list. Seemed the right thing to do. I hope this made it more better than worse. :)
 * I had the idea that maybe a column should be added named "Distribution" that could have values like "TV special" or "direct-to-video". 80.98.191.91 (talk) 13:15, 20 January 2024 (UTC)

title: Talk:List of Babylon 5 episodes


text:
Hatherleigh

Hatherleigh is a small market town in west Devon, England.

It hosts an arts festival in July, and a carnival in November featuring two flaming tar barrel runs. The Walruses meet on New Year's Day to jump into the River Lew to raise money for local good causes, this tradition which started in the 1980s came to end on January 1, 2018. An annual half-marathon called the Ruby Run takes place usually in June between Holsworthy and Hatherleigh, starting from each town in alternate years.

Hatherleigh Market has weekly sales of sheep, cattle and poultry with increased sales on Tuesdays. It is the smallest town in Devon. The market formally closed in February 2018 due to the site being sold to make way for a new housing development, the Tuesday pannier market continues and there will be provision and a new building in the development. In September 2019 the demolition of the site began as of March 2020 only one building of the market remains to temporary house the pannier market, the new development is moving at pace with many of the new buildings now rising from the ground.

Hatherleigh is home to two pubs, The Tally Ho and The George. The George was burned down in an arson attack on 23 December 2008 and has subsequently been rebuilt and re-opened in 2010. More than 100 firefighters from across the county were drafted in to help fight the blaze which was still burning well into the early hours of Christmas Eve. Hatherleigh also has a cafe, two hairdressers, craft shops, a beauty salon, a butcher, a newsagency, a vets, medical centre, a convenience shop and a petrol station.

The parish church is dedicated to St John the Baptist and has a western tower topped by a shingled spire. The spire was destroyed in the Burns' Day Storm of 1990 and a replica of the original now stands in its place. The nave and aisles are divided by granite arcades. The font is Norman and the original wagon roofs remain.

An impressive obelisk one mile (1.6 km) east of the town commemorates Lt Col. William Morris (d. 1858 in India): it was built in 1860.

Hatherleigh was the beginning of the 7th stage of the Tour of Britain in September 2009 and record numbers turned out to watch the famous race begin. Hatherleigh had often been on the course of the Tour but this was the first time a stage had started in the town.

Hatherleigh saw the Olympic torch pass through the town on 21 May 2012.

There is an electoral ward with the same name. The population at the 2011 census was 2,218.

There is no railway station within Hatherleigh's boundaries. However residents can use the nearby railway station in Okehampton now that the Dartmoor Line has reopened with regular services to Exeter.

The village is part of the Ruby Country which covers 45 parishes around the market towns of Holsworthy and Hatherleigh. These two towns were at the centre of the 2001 Foot and Mouth Disease outbreak, and although agriculture was directly affected, most local businesses suffered considerable financial hardship. As a result, the Ruby Country Initiative was established, a not for profit partnership, to help create a more robust and sustainable local economy, and to create an identity for the area.

title: Hatherleigh


text:
Double majority

A double majority is a voting system which requires a majority of votes according to two separate criteria. The mechanism is usually used to require strong support for any measure considered to be of great importance. Typically in legislative bodies, a double majority requirement exists in the form of a quorum being necessary for legislation to be passed.

Australia
In Australia, constitutional changes must be passed at a referendum in a majority of states (4 of the 6), and by a majority of voters nationally. Prior to 1977, the votes of citizens in the Northern Territory and the ACT did not affect the national or state-based count. After a Constitution Alteration put to referendum in 1977 and given vice-regal assent on 19 July 1977, Territorial votes contribute towards the national majority, but the Territories themselves do not count towards the majority of states. An earlier referendum in 1974, where the same proposed change had been bundled with a lowering of the required number of states to a half (3 of the 6) instead of a majority, had been defeated.

Canada
Since the patriation of the Canadian constitution in 1982, thorough amending formulae for the constitution were adopted. Per the Constitution Act, 1982, many amendments can be passed only by the Parliament of Canada and a two-thirds majority of the provincial legislatures, those provinces together representing at least 50% of the national population-–this is known as the 7/50 formula (as there were and are 10 provinces, so 7 constitutes a two-thirds majority). Additionally, a province can explicitly choose to dissent to an amendment that "derogates from the legislative powers, the proprietary rights or any other rights or privileges of the legislature or government of a province", in which case it does not apply in that province even if passed. Though not constitutionally mandated, a referendum is also considered to be necessary by some, especially following the precedent established by the Charlottetown Accord in 1992.

However, there are some parts of the constitution that can be modified only by a vote of all the provinces plus the Parliament of Canada; these include changes to the composition of the Supreme Court of Canada, changing the process for amending the constitution itself, or any act affecting the Canadian monarch or Governor General.

European Union
In the European Union, double majority voting replaced artificial voting weights for votes requiring a qualified majority in the Council of the European Union following implementation of the Treaty of Lisbon. A qualified majority requires 55% of voting EU member states, representing at least 65% of the population of voting members for a European Commission proposal to be approved. This increases to 72% of voting members states, representing at least 65% of the EU population of voting members should the proposal originate from a member state. Proposals can be blocked should a qualified majority of at least four Council members representing more than 35% of the EU population be formed.

Finland
Changing the constitution of Finland requires that a simple majority of the sitting Parliament vote in favor of the amendment. The amendment is postponed until the next general election. The next parliament may finally vote to ratify the amendment but by a two thirds of the MPs. Thus, a double majority of two different parliaments is usually required to pass constitutional amendments. An expedited process may however be entered if five sixths of the sitting parliament vote to declare an amendment urgent. Then, a two-thirds majority of the current parliament may ratify an amendment using the expedited process.

Northern Ireland
Under the terms of the Belfast Agreement, if 30 members or more request it, a measure may be put to a "cross-community vote" which requires a majority from both the Nationalist and Unionist camps.

Philippines
In the 2019 Bangsamoro autonomy plebiscite, for a place to join the Bangsamoro, it is required to have a majority for inclusion both in the place that is petitioning to be included, and from the rest of the larger entity it is originally included. For example, for the six towns from Lanao del Norte petitioning to join, a majority both from the town itself and from the rest of Lanao del Norte should be attained. Each of the six towns overwhelmingly voted for inclusion, but a majority from the rest of Lanao del Norte rejected the towns joining the Bangsamoro, leading to the towns not being included. Meanwhile, 63 of the 67 barangays (villages) in Cotabato voted for inclusion and got consent from their mother towns to join. One barangay rejected inclusion but was allowed by its mother town to join, while three barangays voted for inclusion but were rejected by their mother towns from joining; these four barangays did not join the Bangsamoro. Finally, Isabela City rejected to join, but was allowed by its mother province Basilan to join; it was also not included in the Bangsamoro.

Romania
In Romania, a national referendum is considered valid only if at least 50% plus one of the registered voters cast their ballot. For a valid referendum, the outcome is determined by a simple majority of valid votes cast. The whole process thus involves a double majority in form of a quorum.

A controversial amendment to the referendum law arose during the 2012 political crisis and allowed for an exception to be made in case of a referendum regarding the impeachment of the President. The original law, which required a double majority for impeachment, was modified by governmental emergency decree to circumvent the quorum requirement. This would have resulted in a successful impeachment of the President in the following referendum. The referendum was, however, subsequently invalidated by the Constitutional Court, the body responsible for overseeing the whole process, putting an end to the crisis.

Sudan
The independence referendum for South Sudan required 51% of the vote and 60% turnout.

Switzerland
In Switzerland, the passing of a constitutional amendment by popular vote requires a double majority; not only must a majority of people vote for the amendment but a majority of cantons must also give their consent. This is to prevent a larger canton from foisting amendments onto the smaller ones and vice versa.

United States
Double majority is used in the United States for some initiative or referendum votes on issues such as a tax levy or bond. Essentially, a double majority standard applies a two-part test to a vote outcome before a measure is passed: If the answer to either of these questions is No, the measure fails.
 * Did a majority of registered voters turn out for the election?
 * Did votes in favor outnumber votes against, by the required margin?

This mechanism is used to prevent a small group from passing spending measures that affect the entire population in order to support their pet causes, especially at an election expected to have low voter turnout. Double majorities are also frequently used in municipal annexations, wherein majorities of both the residents in the annexing territory and the territory to be annexed must support the annexation. A similar rule exists for adopting metropolitan government in Tennessee, where the referendum must pass both inside and outside the principal city.

title: Double majority


text:
Peters Marland

Peters Marland is a small village and civil parish in the local government district of Torridge, Devon, England. The parish, which lies about four miles south of the town of Great Torrington, is surrounded clockwise from the north by the parishes of Little Torrington, Merton, Petrockstowe, Buckland Filleigh, Shebbear and Langtree. In 2001 its population was 234, down from the 286 residents it had in 1901.

In 1850 the parish was recorded as covering 2,200 acres with 351 parishioners. At that time most of the land within the parish belonged to Rev. John Moore-Stevens (died 1865), Archdeacon of Exeter, whose son was living at Winscott House in the parish; much also belonged to G. Oldham of Twigbear. Both Winscott and Twigbear are former manors that have their origins before the Norman Conquest, as also are Week and Winswell in the parish.

The parish church, in the village, is dedicated to St Peter. It was extensively restored in the 1860s by the Moore-Stevens family and is, according to W. G. Hoskins, "entirely without interest".

Industry
Ball clay is quarried in the east of the parish, as it has been for many years. There was a brick and tile works here until 1940; many houses in Great Torrington are built of its cream-coloured bricks. The North Devon and Cornwall Junction Light Railway served the works between 1925 and 1982. Today the former railway line forms part of the Tarka Trail series of footpaths and cycle tracks.

title: Peters Marland


text:
Bryn Celli Ddu

Bryn Celli Ddu is a prehistoric site on the Welsh island of Anglesey located near Llanddaniel Fab. Its name means 'the mound in the dark grove'. It was archaeologically excavated between 1928 and 1929. Visitors can get inside the mound through a stone passage to the burial chamber, and it is the centrepiece of a major Neolithic Scheduled Monument in the care of Cadw. The presence of a mysterious pillar within the burial chamber, the reproduction of the 'Pattern Stone', carved with sinuous serpentine designs, and the fact that the site was once a henge with a stone circle, and may have been used to plot the date of the summer solstice have all attracted much interest.

The monument
Bryn Celli Ddu is generally considered to be one of the finest passage tombs in Wales. Its passage and burial chamber are complete, and it is still buried under a mound or cairn, reinstated following its excavation in 1929. (Many stone chambered cairns have lost these features.) As it now stands, the passage is 8.4 m long, the first 3.4 m being unroofed with a pair of portal stones. The main passage has walls of vertical rock slabs, roofed by a series of stone lintels. The mound, being substantially smaller than as originally made, no longer completely encloses the burial chamber, so the back wall is open to the air, allowing some natural light in.

Free-standing inside the burial chamber is a smooth pillar of blueschist, a metamorphic rock, some 2 m high, with a very rounded shape.

Beyond the back wall of the chamber, in a location that would once have been within the mound, is a replica of the 'Pattern Stone'. This was found buried under the mound, and has been put standing up in what is thought to have been its original location at a time when the site was a henge rather than a tomb. The patterns take the form of sinuous serpentine shapes that wind around both sides of the stone. Inside the tomb another stone has a small spiral pattern chipped into it, although its authenticity has been questioned.

Outside the tomb, a ring of kerbstones shows the original extent of the mound, and they also follow the line of the ditch of the earlier henge monument. Three of the stones, visible within the cairn mound, are thought to be from the stone circle of that time.

The passage is roughly aligned with the Summer Solstice sunrise, such that for some weeks around the summer solstice, sunlight can find its way through to the back wall of the burial chamber.

The monument is part of a cluster of Neolithic and Bronze Age features. Two further cairns have been identified just to the south of Bryn Celli Ddu, while in the field immediately to the west is a standing stone, and a rock outcrop with cupmarks carved into it.

Original uses
The earliest identified remains at the site are a row of five postholes previously thought to have been contemporary with the tomb. Radiocarbon dating of pine charcoal from two of the pits, carried out in 2006, showed these to date from around 4000 BC, putting them at the end of the Mesolithic, 1,000 years before the next phase of use. Their purpose, however, is unknown.

Henge
In around 3000 BC a henge monument was constructed. An outer circular bank and ditch would have defined the boundary, although only the ditch survives, some 21 m across. Within this, a stone circle would have provided the focus for a site of ritual significance. A ring of 17 stones formed an oval, many being in matched pairs either side of the centre. Cremated human remains were buried at the base of some of them, suggesting a central 'altar' During this period a pit was dug within the henge; a single human ear-bone was buried, and covered with a flat slab. A second stone known as 'The Pattern Stone' lay nearby, with the serpentine patterning on it. It is thought this would have stood upright within the henge, as the patterns cover both sides.

Passage grave
Some 1000 years after the henge was built, the site was radically altered. All but one of the standing stones were intentionally damaged, some were knocked over and six were smashed with heavy stones. In its place a passage grave was built. Much larger than the mound now remaining, it would have had a complete circle of kerbstones following the line of the old henge ditch, creating an impressive retaining wall around the mound, 26 m across. The burial chamber would have been entirely enclosed within the mound, rather than the back wall being open to the air, as in the reconstructed mound now seen. Individual human bones, both burnt and unburnt, were found within the chamber and passage, suggesting a variety of funeral practices, but in all cases re-using the tomb, and clearing aside the old remains.

At the end of its period of use the tomb was 'closed' by means of a large stone set across the entrance, between the two portal stones.

Archaeology
The earliest archaeological descriptions date to around 1800. In 1796 it was included on a list of Anglesey cromlechs in the Cambrian Register. In 1802 Rev John Skinner made a "Ten Days' Tour Through the Isle of Anglesea", an account of which he wrote up, but never published, describing the numerous archaeological sites he visited. It was finally published as a supplement to Archaeologia Cambrensis in 1908. On his visit to Bryn Celli Ddu, he was told how the passage tomb had been discovered a generation before, by a farmer looking for useful stone. The sight of the pillar stone had initially unnerved the discoverer, but the prospect of treasure tempted him back, and he uprooted the pillar. Having suffered depredation of the more movable stones of the site, the monument was excavated by W J Hemp in 1928–29. He revealed much of the sequence of use on the site, and found the 'Pattern Stone'. It has since been moved to the National Museum of Wales and replaced with a replica standing outside the back wall, but within the area of the original mound, and close to the spot where it was found. The Pillar was set back upright, and the earth and stone mound covering the passage and chamber was reinstated, but clearly there was much less material available than had been used originally. Some additional concrete roof supports have also been added at some point.

Norman Lockyer, who in 1906 published the first systematic study of megalithic astronomy, had argued that Bryn Celli Ddu marked the summer solstice. This was ridiculed at the time, but research by Christopher Knight and Robert Lomas in 1997- 98 showed it to be true. Knight and Lomas also claimed that year-round alignments allowed the site to be used as an agricultural calendar. Steve Burrow, curator of Neolithic archaeology at Amgueddfa Cymru (National Museum of Wales), has more recently supported the case for summer solstice alignment. This alignment links Bryn Celli Ddu to a handful of other sites, including Maes Howe, Orkney and Newgrange, Ireland, both of which point to the winter solstice. It has also been suggested that a feature similar to the 'lightbox' at Newgrange may be matched at Bryn Celli Ddu.

Multi-period landscape
Between 2015 and 2020 the 'Bryn Celli Ddu Public Archaeology Project' has been engaged in studies of the wider landscape around the site. This is a partnership between Cadw and Manchester Metropolitan University with the aim of involving local communities in archaeological investigation. This has resulted in study of known ancient rock art on a nearby outcrop, and also identified and uncovered a number of other rock outcrops close by that have 'cup mark' decorations chipped into them. A second burial chamber, 50m to the south of the extant one, had been leveled, but as part of the project a 2019 excavation has revealed a large circular burial cairn which radio-carbon dating has placed at 1900 BC, which is over a thousand years after the earlier passage grave was built.

Media
The serpentine pattern and the passage tomb featured strongly in the short animated film, Songs from Stones, about some of Anglesey's evocative archaeological sites and artefacts, produced as part of the Cultural Olympiad in 2012.

The HeritageTogether project has used photogrammetry to create 3D models of the site and the standing stone.

Media company Mint Motion have produced an animated video of the site's development over time, and a flythough of a pointcloud model of the site (see external links below).

The site was featured and explored in an episode of the US reality television series Expedition Unknown.

title: Bryn Celli Ddu


text:
Fracture mechanics

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be used to describe the state around a crack. Fracture mechanics is used to characterise the loads on a crack, typically using a single parameter to describe the complete loading state at the crack tip. A number of different parameters have been developed. When the plastic zone at the tip of the crack is small relative to the crack length the stress state at the crack tip is the result of elastic forces within the material and is termed linear elastic fracture mechanics (LEFM) and can be characterised using the stress intensity factor $$K$$. Although the load on a crack can be arbitrary, in 1957 G. Irwin found any state could be reduced to a combination of three independent stress intensity factors:
 * Mode I – Opening mode (a tensile stress normal to the plane of the crack),
 * Mode II – Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front), and
 * Mode III – Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front).

When the size of the plastic zone at the crack tip is too large, elastic-plastic fracture mechanics can be used with parameters such as the J-integral or the crack tip opening displacement.

The characterising parameter describes the state of the crack tip which can then be related to experimental conditions to ensure similitude. Crack growth occurs when the parameters typically exceed certain critical values. Corrosion may cause a crack to slowly grow when the stress corrosion stress intensity threshold is exceeded. Similarly, small flaws may result in crack growth when subjected to cyclic loading. Known as fatigue, it was found that for long cracks, the rate of growth is largely governed by the range of the stress intensity $$\Delta K$$ experienced by the crack due to the applied loading. Fast fracture will occur when the stress intensity exceeds the fracture toughness of the material. The prediction of crack growth is at the heart of the damage tolerance mechanical design discipline.

Motivation
The processes of material manufacture, processing, machining, and forming may introduce flaws in a finished mechanical component. Arising from the manufacturing process, interior and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture mechanics is the analysis of flaws to discover those that are safe (that is, do not grow) and those that are liable to propagate as cracks and so cause failure of the flawed structure. Despite these inherent flaws, it is possible to achieve through damage tolerance analysis the safe operation of a structure. Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new.

Fracture mechanics should attempt to provide quantitative answers to the following questions:

 * 1) What is the strength of the component as a function of crack size?
 * 2) What crack size can be tolerated under service loading, i.e. what is the maximum permissible crack size?
 * 3) How long does it take for a crack to grow from a certain initial size, for example the minimum detectable crack size, to the maximum permissible crack size?
 * 4) What is the service life of a structure when a certain pre-existing flaw size (e.g. a manufacturing defect) is assumed to exist?
 * 5) During the period available for crack detection how often should the structure be inspected for cracks?

Griffith's criterion

Fracture mechanics was developed during World War I by English aeronautical engineer A. A. Griffith – thus the term Griffith crack – to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts:

 * The stress needed to fracture bulk glass is around 100 MPa.
 * The theoretical stress needed for breaking atomic bonds of glass is approximately 10000 MPa.

A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.

To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental glass specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length ($$a$$) and the stress at fracture ($$\sigma_f$$) was nearly constant, which is expressed by the equation:

 * $$\sigma_f\sqrt{a} \approx C$$

An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed.

The growth of a crack, the extension of the surfaces on either side of the crack, requires an increase in the surface energy. Griffith found an expression for the constant $$C$$ in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was:

 * Compute the potential energy stored in a perfect specimen under a uniaxial tensile load.
 * Fix the boundary so that the applied load does no work and then introduce a crack into the specimen. The crack relaxes the stress and hence reduces the elastic energy near the crack faces. On the other hand, the crack increases the total surface energy of the specimen.
 * Compute the change in the free energy (surface energy − elastic energy) as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases as the crack length increases, i.e. by causing fracture. Using this procedure, Griffith found that

 * $$C = \sqrt{\cfrac{2E\gamma}{\pi}}$$

where $$E$$ is the Young's modulus of the material and $$\gamma$$ is the surface energy density of the material. Assuming $$E = 62\ \text{GPa}$$ and $$\gamma = 1\ \text{J/m}^2$$ gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass.

For the simple case of a thin rectangular plate with a crack perpendicular to the load, the energy release rate, $$G$$, becomes:

 * $$G = \frac{\pi \sigma^2 a}{E}\,$$

where $$\sigma$$ is the applied stress, $$a$$ is half the crack length, and $$E$$ is the Young's modulus, which for the case of plane strain should be divided by the plate stiffness factor $$(1-\nu^2)$$. The strain energy release rate can physically be understood as: the rate at which energy is absorbed by growth of the crack.

However, we also have that:

 * $$G_c = \frac{\pi \sigma_f^2 a}{E}\,$$

If $$G$$ ≥ $$G_c$$, this is the criterion for which the crack will begin to propagate.

For materials highly deformed before crack propagation, the linear elastic fracture mechanics formulation is no longer applicable and an adapted model is necessary to describe the stress and displacement field close to crack tip, such as on fracture of soft materials.

Irwin's modification

''Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic.''

Griffith's theory provides excellent agreement with experimental data for brittle materials such as glass. For ductile materials such as steel, although the relation $$ \sigma_f\sqrt{a} = C $$ still holds, the surface energy (γ) predicted by Griffith's theory is usually unrealistically high. A group working under G. R. Irwin at the U.S. Naval Research Laboratory (NRL) during World War II realized that plasticity must play a significant role in the fracture of ductile materials.

In ductile materials (and even in materials that appear to be brittle ), a plastic zone develops at the tip of the crack. As the applied load increases, the plastic zone increases in size until the crack grows and the elastically strained material behind the crack tip unloads. The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat. Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy is needed for crack growth in ductile materials as compared to brittle materials.

Irwin's strategy was to partition the energy into two parts: Then the total energy is:
 * the stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for fracture.
 * the dissipated energy which includes plastic dissipation and the surface energy (and any other dissipative forces that may be at work). The dissipated energy provides the thermodynamic resistance to fracture.
 * $$G = 2\gamma + G_p$$

where $$\gamma$$ is the surface energy and $$G_p$$ is the plastic dissipation (and dissipation from other sources) per unit area of crack growth.

The modified version of Griffith's energy criterion can then be written as

 * $$\sigma_f\sqrt{a} = \sqrt{\cfrac{E~G}{\pi}}.$$

For brittle materials such as glass, the surface energy term dominates and $$G \approx 2\gamma = 2 \,\, \text{J/m}^2$$. For ductile materials such as steel, the plastic dissipation term dominates and $$G \approx G_p = 1000 \,\, \text{J/m}^2$$. For polymers close to the glass transition temperature, we have intermediate values of $$G$$ between 2 and 1000 $$\text{J/m}^2$$.

Stress intensity factor
Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid. This asymptotic expression for the stress field in mode I loading is related to the stress intensity factor $$ K_I $$ following:

 * $$\sigma_{ij} = \left(\cfrac{K_{I}}{\sqrt{2\pi r}}\right)~f_{ij}(\theta)$$

where $$ \sigma_{ij} $$ are the Cauchy stresses, $$ r $$ is the distance from the crack tip, $$ \theta $$ is the angle with respect to the plane of the crack, and $$ f_{ij} $$ are functions that depend on the crack geometry and loading conditions. Irwin called the quantity $$ K $$ the stress intensity factor. Since the quantity $$ f_{ij} $$ is dimensionless, the stress intensity factor can be expressed in units of $$\text{MPa}\sqrt{\text{m}}$$.

Stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface weakness energy. Both of these terms are simply related to the energy terms that Griffith used:

 * $$K_I = \sigma \sqrt{\pi a}\,$$

and

$$  K_c = \begin{cases} \sqrt{EG_c} & \text{for plane stress} \\ \\                    \sqrt{\cfrac{EG_c}{1-\nu^2}} & \text{for plane strain} \end{cases} $$

where $$K_I$$ is the mode $$ I $$ stress intensity, $$K_c$$ the fracture toughness, and $$\nu$$ is Poisson’s ratio.

Fracture occurs when $$K_I \geq K_c$$. For the special case of plane strain deformation, $$K_c$$ becomes $$K_{Ic}$$ and is considered a material property. The subscript $$ I $$ arises because of the different ways of loading a material to enable a crack to propagate. It refers to so-called "mode $$  I $$" loading as opposed to mode $$ II $$ or $$ III $$:

The expression for $$K_I$$ will be different for geometries other than the center-cracked infinite plate, as discussed in the article on the stress intensity factor. Consequently, it is necessary to introduce a dimensionless correction factor, $$ Y $$, in order to characterize the geometry. This correction factor, also often referred to as the geometric shape factor, is given by empirically determined series and accounts for the type and geometry of the crack or notch. We thus have:

 * $$K_I = Y \sigma \sqrt{\pi a}\,$$

where $$ Y $$ is a function of the crack length and width of sheet given, for a sheet of finite width $$ W $$ containing a through-thickness crack of length $$ 2a $$, by:

 * $$Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\,$$

Strain energy release
Irwin was the first to observe that if the size of the plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress (the plastic zone) at the crack tip. In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture.

The energy release rate for crack growth or strain energy release rate may then be calculated as the change in elastic strain energy per unit area of crack growth, i.e.,

 * $$G := \left[\cfrac{\partial U}{\partial a}\right]_P = -\left[\cfrac{\partial U}{\partial a}\right]_u$$

where U is the elastic energy of the system and a is the crack length. Either the load P or the displacement u are constant while evaluating the above expressions.

Irwin showed that for a mode I crack (opening mode) the strain energy release rate and the stress intensity factor are related by:

G = G_I = \begin{cases} \cfrac{K_I^2}{E} & \text{plane stress} \\ \cfrac{(1-\nu^2) K_I^2}{E} & \text{plane strain} \end{cases} $$ where E is the Young's modulus, ν is Poisson's ratio, and KI is the stress intensity factor in mode I. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode I, mode II (sliding mode), and mode III (tearing mode) stress intensity factors for the most general loading conditions.

Next, Irwin adopted the additional assumption that the size and shape of the energy dissipation zone remains approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material. This new material property was given the name fracture toughness and designated GIc. Today, it is the critical stress intensity factor KIc, found in the plane strain condition, which is accepted as the defining property in linear elastic fracture mechanics.

Crack tip plastic zone
In theory the stress at the crack tip where the radius is nearly zero, would tend to infinity. This would be considered a stress singularity, which is not possible in real-world applications. For this reason, in numerical studies in the field of fracture mechanics, it is often appropriate to represent cracks as round tipped notches, with a geometry dependent region of stress concentration replacing the crack-tip singularity. In actuality, the stress concentration at the tip of a crack within real materials has been found to have a finite value but larger than the nominal stress applied to the specimen.

Nevertheless, there must be some sort of mechanism or property of the material that prevents such a crack from propagating spontaneously. The assumption is, the plastic deformation at the crack tip effectively blunts the crack tip. This deformation depends primarily on the applied stress in the applicable direction (in most cases, this is the y-direction of a regular Cartesian coordinate system), the crack length, and the geometry of the specimen. To estimate how this plastic deformation zone extended from the crack tip, Irwin equated the yield strength of the material to the far-field stresses of the y-direction along the crack (x direction) and solved for the effective radius. From this relationship, and assuming that the crack is loaded to the critical stress intensity factor, Irwin developed the following expression for the idealized radius of the zone of plastic deformation at the crack tip:
 * $$r_p = \frac{K_{C}^2}{2\pi\sigma_Y^2}$$

Models of ideal materials have shown that this zone of plasticity is centered at the crack tip. This equation gives the approximate ideal radius of the plastic zone deformation beyond the crack tip, which is useful to many structural scientists because it gives a good estimate of how the material behaves when subjected to stress. In the above equation, the parameters of the stress intensity factor and indicator of material toughness, $$K_C$$, and the yield stress, $$\sigma_Y$$, are of importance because they illustrate many things about the material and its properties, as well as about the plastic zone size. For example, if $$K_c$$ is high, then it can be deduced that the material is tough, and if $$\sigma_Y$$ is low, one knows that the material is more ductile. The ratio of these two parameters is important to the radius of the plastic zone. For instance, if $$\sigma_Y$$ is small, then the squared ratio of $$K_C$$ to $$\sigma_Y$$ is large, which results in a larger plastic radius. This implies that the material can plastically deform, and, therefore, is tough. This estimate of the size of the plastic zone beyond the crack tip can then be used to more accurately analyze how a material will behave in the presence of a crack.

The same process as described above for a single event loading also applies and to cyclic loading. If a crack is present in a specimen that undergoes cyclic loading, the specimen will plastically deform at the crack tip and delay the crack growth. In the event of an overload or excursion, this model changes slightly to accommodate the sudden increase in stress from that which the material previously experienced. At a sufficiently high load (overload), the crack grows out of the plastic zone that contained it and leaves behind the pocket of the original plastic deformation. Now, assuming that the overload stress is not sufficiently high as to completely fracture the specimen, the crack will undergo further plastic deformation around the new crack tip, enlarging the zone of residual plastic stresses. This process further toughens and prolongs the life of the material because the new plastic zone is larger than what it would be under the usual stress conditions. This allows the material to undergo more cycles of loading. This idea can be illustrated further by the graph of Aluminum with a center crack undergoing overloading events.

Limitations

But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a crack. One basic assumption in Irwin's linear elastic fracture mechanics is small scale yielding, the condition that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures.

Linear-elastic fracture mechanics is of limited practical use for structural steels and Fracture toughness testing can be expensive.

Elastic–plastic fracture mechanics

Most engineering materials show some nonlinear elastic and inelastic behavior under operating conditions that involve large loads. In such materials the assumptions of linear elastic fracture mechanics may not hold, that is,
 * the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size
 * the size and shape of the plastic zone may change as the applied load is increased and also as the crack length increases.

Therefore, a more general theory of crack growth is needed for elastic-plastic materials that can account for:
 * the local conditions for initial crack growth which include the nucleation, growth, and coalescence of voids (decohesion) at a crack tip.
 * a global energy balance criterion for further crack growth and unstable fracture.

CTOD
Historically, the first parameter for the determination of fracture toughness in the elasto-plastic region was the crack tip opening displacement (CTOD) or "opening at the apex of the crack" indicated. This parameter was determined by Wells during the studies of structural steels, which due to the high toughness could not be characterized with the linear elastic fracture mechanics model. He noted that, before the fracture happened, the walls of the crack were leaving and that the crack tip, after fracture, ranged from acute to rounded off due to plastic deformation. In addition, the rounding of the crack tip was more pronounced in steels with superior toughness.

There are a number of alternative definitions of CTOD. In the two most common definitions, CTOD is the displacement at the original crack tip and the 90 degree intercept. The latter definition was suggested by Rice and is commonly used to infer CTOD in finite element models of such. Note that these two definitions are equivalent if the crack tip blunts in a semicircle.

Most laboratory measurements of CTOD have been made on edge-cracked specimens loaded in three-point bending. Early experiments used a flat paddle-shaped gage that was inserted into the crack; as the crack opened, the paddle gage rotated, and an electronic signal was sent to an x-y plotter. This method was inaccurate, however, because it was difficult to reach the crack tip with the paddle gage. Today, the displacement V at the crack mouth is measured, and the CTOD is inferred by assuming the specimen halves are rigid and rotate about a hinge point (the crack tip).

R-curve
An early attempt in the direction of elastic-plastic fracture mechanics was Irwin's crack extension resistance curve, Crack growth resistance curve or R-curve. This curve acknowledges the fact that the resistance to fracture increases with growing crack size in elastic-plastic materials. The R-curve is a plot of the total energy dissipation rate as a function of the crack size and can be used to examine the processes of slow stable crack growth and unstable fracture. However, the R-curve was not widely used in applications until the early 1970s. The main reasons appear to be that the R-curve depends on the geometry of the specimen and the crack driving force may be difficult to calculate.

J-integral
In the mid-1960s James R. Rice (then at Brown University) and G. P. Cherepanov independently developed a new toughness measure to describe the case where there is sufficient crack-tip deformation that the part no longer obeys the linear-elastic approximation. Rice's analysis, which assumes non-linear elastic (or monotonic deformation theory plastic) deformation ahead of the crack tip, is designated the J-integral. This analysis is limited to situations where plastic deformation at the crack tip does not extend to the furthest edge of the loaded part. It also demands that the assumed non-linear elastic behavior of the material is a reasonable approximation in shape and magnitude to the real material's load response. The elastic-plastic failure parameter is designated JIc and is conventionally converted to KIc using the equation below. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior.

The mathematical definition of J-integral is as follows:

J= \int_\Gamma( w \,dy - T_i \frac{\partial u_i}{\partial x}\,ds) \quad \text{with} \quad w=\int^{\varepsilon_{ij}}_0 \sigma_{ij} \,d\varepsilon_{ij} $$

where
 * $$\Gamma$$ is an arbitrary path clockwise around the apex of the crack,
 * $$w$$ is the density of strain energy,
 * $$T_i$$ are the components of the vectors of traction,
 * $$u_i$$ are the components of the displacement vectors,
 * $$ds$$ is an incremental length along the path $$\Gamma$$, and
 * $$\sigma_{ij}$$ and $$\varepsilon_{ij}$$ are the stress and strain tensors.

Since engineers became accustomed to using KIc to characterise fracture toughness, a relation has been used to reduce JIc to it:
 * $$K_{Ic} = \sqrt{E^* J_{Ic}}\,$$ where $$E^* = E$$ for plane stress and $$E^* = \frac{E}{1 - \nu^2}$$ for plane strain.

Cohesive zone model
When a significant region around a crack tip has undergone plastic deformation, other approaches can be used to determine the possibility of further crack extension and the direction of crack growth and branching. A simple technique that is easily incorporated into numerical calculations is the cohesive zone model method which is based on concepts proposed independently by Barenblatt and Dugdale in the early 1960s. The relationship between the Dugdale-Barenblatt models and Griffith's theory was first discussed by Willis in 1967. The equivalence of the two approaches in the context of brittle fracture was shown by Rice in 1968.

Transition flaw size

Let a material have a yield strength $$\sigma_Y$$ and a fracture toughness in mode I $$K_{Ic}$$. Based on fracture mechanics, the material will fail at stress $$\sigma_\text{fail}=K_{Ic}/\sqrt{\pi a}$$. Based on plasticity, the material will yield when $$\sigma_{fail}=\sigma_Y$$. These curves intersect when $$a=K_{Ic}^2/\pi\sigma_Y^2$$. This value of $$a$$ is called as transition flaw size $$ a_t$$., and depends on the material properties of the structure. When the $$aa_t $$ the failure is governed by fracture mechanics. The value of $$a_t$$ for engineering alloys is 100 mm and for ceramics is 0.001 mm. If we assume that manufacturing processes can give rise to flaws in the order of micrometers, then, it can be seen that ceramics are more likely to fail by fracture, whereas engineering alloys would fail by plastic deformation.

Concrete fracture analysis
Concrete fracture analysis is part of fracture mechanics that studies crack propagation and related failure modes in concrete. As it is widely used in construction, fracture analysis and modes of reinforcement are an important part of the study of concrete, and different concretes are characterized in part by their fracture properties. Common fractures include the cone-shaped fractures that form around anchors under tensile strength.

Bažant (1983) proposed a crack band model for materials like concrete whose homogeneous nature changes randomly over a certain range. He also observed that in plain concrete, the size effect has a strong influence on the critical stress intensity factor, and proposed the relation $$\sigma$$ = $$\tau$$ / √(1+{$$d$$/$$\lambda$$$$\delta$$}), where $$\sigma$$ = stress intensity factor, $$\tau$$ = tensile strength, $$d$$ = size of specimen, $$\delta$$ = maximum aggregate size, and $$\lambda$$ = an empirical constant.

Atomistic Fracture Mechanics
Atomistic Fracture Mechanics (AFM) is a relatively new field that studies the behavior and properties of materials at the atomic scale when subjected to fracture. It integrates concepts from fracture mechanics with atomistic simulations to understand how cracks initiate, propagate, and interact with the microstructure of materials. By using techniques like Molecular Dynamics (MD) simulations, AFM can provide insights into the fundamental mechanisms of crack formation and growth, the role of atomic bonds, and the influence of material defects and impurities on fracture behavior.

title: Fracture mechanics


text:
Marine Corps Brevet Medal

The Marine Corps Brevet Medal, also known as the Brevet Medal, was a military decoration of the United States Marine Corps; it was created in 1921 as a result of Marine Corps Order Number 26. The decoration was a one-time issuance and retroactively recognized living Marine Corps officers who had received a brevet rank. The similar practice of frocking continues in all six branches of the U.S. Armed Forces.

Brevet promotions were used by the United States military in some capacity from 1775 until they were discontinued in 1900. The Army was the only branch authorized to grant brevets until 1814, when the Marine Corps was granted the same privilege. Over 86 years, the Marine Corps awarded 121 brevet promotions to 100 Marine Corps officers. Captain Anthony Gale was the first to receive a brevet promotion in 1814, and John Twiggs Myers, who died in 1952, was the last surviving recipient.

In 1921, Commandant John A. Lejeune requested that a Marine Corps Brevet Medal be authorized. After it was approved and created, the decoration was given to the last 20 living Marine Corps officers who received brevet promotions.

Brevet promotions
A brevet promotion, or brevet, is the advancement in rank without the advancement in either pay grade or position. Typically, a brevetted officer would be given the insignia of the brevetted rank but not the pay or formal authority. Brevet promotions were originally authorized for the United States Army in 1775 by the Second Continental Congress. In 1778, a resolution was passed stating brevets would only be authorized "..to officers in the line or in case of very eminent services...".

The Marine Corps would not receive the authorization from Congress for brevet promotions until 1814, stating "... That the President is hereby authorized to confer brevet rank on such officers of the Marine Corps as shall distinguish themselves by gallant actions and meritorious conduct or shall have served ten years in any one grade...".

In 1814 Anthony Gale became the first Marine to receive a brevet promotion when he was brevetted to Major, having been a Captain for ten years. By the time the practice of brevet promotions was discontinued in 1900, 121 brevet promotions were bestowed on 100 Marine Corps officers.

In the 19th and early 20th centuries, brevet promotions were common in the United States armed forces. New officers received brevet rank until authorized positions were made available, or they could be brevetted to fill higher positions for gallantry. During the American Civil War, most senior officers received a brevet promotion, particularly during the final months of the war.

Due to the establishment of the Medal of Honor and the change in rules allowing both officers and enlisted to receive it, the need for brevet promotions diminished. During the American Civil War, the Army used the issuing of brevet promotions to such a degree that Congress passed an act in 1869 that restricted the issuance of brevet promotions. The act established three requirements for awarding a brevet: "they could only be awarded in time of war and then only for distinguished conduct and public service in the presence of the enemy, and it also removed all privileges of command based upon brevet rank except as directed by the President."

In 1870 Congress passed a law stating that no officer could wear, nor be addressed by, their brevet rank making brevet promotions an honorary decoration only. Because of this new law the last nine brevet promotions awarded by the Marine Corps occurred during the Boxer Rebellion. On June 7, 1921, the Secretary of the Navy Edwin Denby approved then-Commandant John A. Lejeune's request for a medal denoting the holder of a brevet promotion to be issued. Marine Corps Order #26 was issued on June 27, 1921, authorizing the medal to be ordered and by November 10, 1921, the medals had been created. This decoration was justified on the grounds that, until 1915, Marine Corps officers were not eligible for the Medal of Honor.

In 1940 the Marine Corps declared the Brevet Medal obsolete; the medal was never issued again. The concept of brevet commissions was phased out of the United States military, and was replaced by temporary and field promotions, which were awarded more frequently than brevet ranks.

Award of the medal was approved for twenty-three men, three of whom died before they could receive this award. Of the twenty recipients, three were holders of both the Marine Corps Brevet Medal and the Medal of Honor.

Description and symbolism
The Marine Corps Brevet Medal was considered to be the equivalent of the Navy Cross, although in precedence it ranks just behind the Medal of Honor. Recipients of the medal had received field commissions as Marine Corps officers, under combat conditions, and had performed feats of distinction and gallant service. Initially, the Brevet Medal ranked behind the Navy Distinguished Service Medal and was issued to 23 active, retired, and discharged Marine Corps personnel.

The medal was designed by Sergeant Joseph Alfred Burnett and contained a ribbon, in USMC scarlet, closely resembling the blue-and-white starred pattern of the Medal of Honor. No attached devices for the Brevet Medal were authorized.

Front of medal
The medal consists of a bronze cross pattée, with the center of each arm extended in a semi-circular shape and in the center of the front is the word "BREVET", encircled by the words "UNITED STATES MARINE CORPS". A small five-pointed star, point-up, is at the bottom center of the circle formed by the inscription and a small Marine Corps insignia (eagle, globe and anchor) attaches the medal to its suspension ring.

Back of medal
The back of the medal is plain except for its center, which contains the inscription "FOR DISTINGUISHED CONDUCT" in a circle, and the words "IN PRESENCE OF ENEMY" in the center. The original medals were neither named nor numbered.

Recipients

 * Approved recipients who died before award of the medal

title: Marine Corps Brevet Medal


text:
Samuel Tertius Galton

Samuel Tertius Galton (23 March 1783 – 23 October 1844) was a British businessman and scientist.

Life
He was the son of Samuel "John" Galton, a prominent member of the scientific Lunar Society, and the father of Francis Galton the eminent Victorian scientist. He was born in the area of Duddeston in Birmingham. Samuel Tertius, though less distinguished, was not an exception to the rule of scientific endeavour in his family.

Galton also inherited his father's considerable business interests and quickly set about making changes—discontinuing the family's armaments business in 1815. Though fascinated by economics (he wrote papers on the subject), Samuel preferred to be less 'hands-on' in the running of the business than his father and spent much of his time living off the revenue of his considerable estate.

Family
Though brought up a Quaker, Samuel converted to the Anglicanism in 1807. On 30 March 1807, he married Violetta (Francis Anne Viollette) Darwin, one of the fourteen children of his father's old colleague and fellow Lunar Society member Erasmus Darwin. Many of the 'Lunar children' grew up together and there were several such marriages. They had four daughters and three sons:

 * Elizabeth Anne Galton (1808–1906), married Edward Wheler
 * Lucy Harriot Galton (1809–1848), married James Moilliet
 * Millicent Adele Galton (1810–1883), married the Rev Robert Shirley Bunbury
 * Emma Sophia Galton (1811–1904)
 * Darwin Galton (1814–1903). High Sheriff of Warwickshire in 1850.
 * Erasmus Galton (1815–1909)
 * Francis Galton (1822–1911)

title: Samuel Tertius Galton


text: Category:Elections in Cambodia

title: Category:Elections in Cambodia


text: Category:Elections in Hong Kong

title: Category:Elections in Hong Kong


text:
User talk:Arcataroger

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User:Picapica

Regarded by many as a thorough pest and a nuisance, Pica pica – the European magpie – is known both for its fondness for collecting bright, glittery objects ( ピカピカ ) and for its "loud, harsh chatter".

Who was that masked bird?

Why, that was the Lone Rearranger...

Ubi et unde
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User:Chmod007/Test

Test multiple lines aoeu abc def

title: User:Chmod007/Test


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User talk:Sheikspeare

Welcome to the Wikipedia
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