Frank-Read Source
Encyclopedia
A Frank-Read Source is a mechanism explaining the generation of multiple dislocation
s in specific well spaced slip
planes in crystal
s when they are deformed. It was proposed by and named after Sir Charles Frank and Thornton Read. When a crystal is deformed, slip is found to occur only on certain well spaced slip planes. Furthermore, it is found that in order for that slip to occur, dislocations must be generated in the material. This implies that, during deformation, dislocations must be primarily generated in these planes. Cold working of metal increases the number of dislocations by the Frank-Read mechanism. Higher dislocation density increases yield strength and causes work hardening
of metals.
in 1980. In 1950 Sir Charles Frank, who was then a research fellow in the physics department at the University of Bristol
, visited the United States to participate in a conference on crystal plasticity
in Pittsburgh. Sir Charles Frank arrived in the United States well in advance of the conference to spend time at a naval laboratory and to give a lecture at Cornell University
. When, during his travels in Pennsylvania, Sir Charles Frank visited Pittsburgh, he received a letter from fellow scientist Jock Eshelby suggesting that he read a recent paper by Gunther Leibfried. Sir Charles Frank was supposed to board a train to Cornell to give his lecture at Cornell, but before departing for Cornell he went to the library at Carnegie Institute of Technology
to obtain a copy of the paper. The library did not yet have the journal with Leibfried's paper, but the staff at the library believed that the journal could be in the recently arrived package from Germany. Sir Charles Frank decided to wait for the library to open the package, which did indeed contain the journal. Upon reading the paper he took a train to Cornell, where he was told to pass the time until 5:00, as the faculty was in meeting. Sir Charles Frank decided to take a walk between 3:00 and 5:00. During those two hours, while considering the Leibfried paper, he formulated the theory for what was later named the Frank-Read source. A couple of days later, he traveled to the conference on crystal plasticity in Pittsburgh where he ran into Thorton Read in the hotel lobby. Upon encountering each other, the two scientists immediately discovered that they had come up with the same idea for dislocation generation almost simultaneously (Sir Charles Frank during his walk at Cornell, and Thorton Read during tea the previous Wednesday) and decided to write a joint paper on the topic. The mechanism for dislocation generation described in that paper is now known as the Frank-Read source. Sir Charles Frank's rendition of this story, on which this summary is based, is a far more engaging read which should be read for a more complete picture.
.
Consider a straight dislocation in a crystal slip plane with its two ends, A and B, pinned. If a shear stress
is exerted on the slip plane then a force 
, where b is the Burgers vector
of the dislocation and x is the distance between the pinning sites A and B, is exerted on the dislocation line as a result of the shear stress. This force acts perpendicular
ly to the line, inducing the dislocation to lengthen and curve into an arc. The bending force caused by the shear stress is opposed by the line tension of the dislocation, which acts on each end of the dislocation along the direction of the dislocation line away from A and B with a magnitude of
, where G is the shear modulus. If the dislocation bends, the ends of the dislocation make an angle with the horizontal between A and B, which gives the line tensions acting along the ends a vertical
component acting directly against the force induced by the shear stress. If sufficient shear stress is applied and the dislocation bends, the vertical component from the line tensions, which acts directly against the force caused by the shear stress, grows as the dislocation approaches a semicircular shape. When the dislocation becomes a semicircle, all of the line tension is acting against the bending force induced by the shear stress, because the line tension is perpendicular to the horizontal
between A and B. For the dislocation to reach this point, it is thus evident that the equation:
must be satisfied, and from this we can solve for the shear stress:
This is the stress required to generate dislocation from a Frank-Read source. If the shear stress increases any further and the dislocation passes the semicircular equilibrium state, it will spontaneously continue to bend and grow, spiraling around the A and B pinning points, until the segments spiraling around the A and B pinning points collide and cancel. The process results in a dislocation loop around A and B in the slip plane which expands under continued shear stress, and also in a new dislocation line between A and B which, under renewed or continued shear, can continue to generate dislocation loops in the manner just described. A Frank-Read loop can thus generate many dislocations in a plane in a crystal under applied stress. The Frank-Read source mechanism explains why dislocations are primarily generated on certain slip planes; dislocations are primarily generated in just those planes with Frank-Read sources. It is important to note that if the shear stress does not exceed:
and the dislocation does not bend past the semicircular equilibrium state, it will not form a dislocation loop and instead revert to its original state.
Dislocation
In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. The presence of dislocations strongly influences many of the properties of materials...
s in specific well spaced slip
Slip (materials science)
In materials science, Slip is the process by which plastic deformation is produced by a dislocation motion. By an external force, parts of the crystal lattice glide along each other, resulting in a changed geometry of the material. Depending on the type of lattice, different slip systems are...
planes in crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
s when they are deformed. It was proposed by and named after Sir Charles Frank and Thornton Read. When a crystal is deformed, slip is found to occur only on certain well spaced slip planes. Furthermore, it is found that in order for that slip to occur, dislocations must be generated in the material. This implies that, during deformation, dislocations must be primarily generated in these planes. Cold working of metal increases the number of dislocations by the Frank-Read mechanism. Higher dislocation density increases yield strength and causes work hardening
Work hardening
Work hardening, also known as strain hardening or cold working, is the strengthening of a metal by plastic deformation. This strengthening occurs because of dislocation movements within the crystal structure of the material. Any material with a reasonably high melting point such as metals and...
of metals.
History
Sir Charles Frank detailed the history of the discovery from his perspective in Proceedings of the Royal SocietyProceedings of the Royal Society
Proceedings of the Royal Society is the parent title of two scientific journals published by the Royal Society, whereas its initial journal, Philosophical Transactions, is now devoted to special thematic issues...
in 1980. In 1950 Sir Charles Frank, who was then a research fellow in the physics department at the University of Bristol
University of Bristol
The University of Bristol is a public research university located in Bristol, United Kingdom. One of the so-called "red brick" universities, it received its Royal Charter in 1909, although its predecessor institution, University College, Bristol, had been in existence since 1876.The University is...
, visited the United States to participate in a conference on crystal plasticity
Plasticity (physics)
In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...
in Pittsburgh. Sir Charles Frank arrived in the United States well in advance of the conference to spend time at a naval laboratory and to give a lecture at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
. When, during his travels in Pennsylvania, Sir Charles Frank visited Pittsburgh, he received a letter from fellow scientist Jock Eshelby suggesting that he read a recent paper by Gunther Leibfried. Sir Charles Frank was supposed to board a train to Cornell to give his lecture at Cornell, but before departing for Cornell he went to the library at Carnegie Institute of Technology
Carnegie Institute of Technology
The Carnegie Institute of Technology , is the name for Carnegie Mellon University’s College of Engineering. It was first called the Carnegie Technical Schools, or Carnegie Tech, when it was founded in 1900 by Andrew Carnegie who intended to build a “first class technical school” in Pittsburgh,...
to obtain a copy of the paper. The library did not yet have the journal with Leibfried's paper, but the staff at the library believed that the journal could be in the recently arrived package from Germany. Sir Charles Frank decided to wait for the library to open the package, which did indeed contain the journal. Upon reading the paper he took a train to Cornell, where he was told to pass the time until 5:00, as the faculty was in meeting. Sir Charles Frank decided to take a walk between 3:00 and 5:00. During those two hours, while considering the Leibfried paper, he formulated the theory for what was later named the Frank-Read source. A couple of days later, he traveled to the conference on crystal plasticity in Pittsburgh where he ran into Thorton Read in the hotel lobby. Upon encountering each other, the two scientists immediately discovered that they had come up with the same idea for dislocation generation almost simultaneously (Sir Charles Frank during his walk at Cornell, and Thorton Read during tea the previous Wednesday) and decided to write a joint paper on the topic. The mechanism for dislocation generation described in that paper is now known as the Frank-Read source. Sir Charles Frank's rendition of this story, on which this summary is based, is a far more engaging read which should be read for a more complete picture.
Mechanism
The Frank-Read Source is a mechanism based on dislocation multiplication in a slip plane under shear stressShear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
.
Consider a straight dislocation in a crystal slip plane with its two ends, A and B, pinned. If a shear stress
is exerted on the slip plane then a force , where b is the Burgers vectorBurgers vector
The Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted b, that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice....
of the dislocation and x is the distance between the pinning sites A and B, is exerted on the dislocation line as a result of the shear stress. This force acts perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
ly to the line, inducing the dislocation to lengthen and curve into an arc. The bending force caused by the shear stress is opposed by the line tension of the dislocation, which acts on each end of the dislocation along the direction of the dislocation line away from A and B with a magnitude of
, where G is the shear modulus. If the dislocation bends, the ends of the dislocation make an angle with the horizontal between A and B, which gives the line tensions acting along the ends a verticalVertical direction
In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...
component acting directly against the force induced by the shear stress. If sufficient shear stress is applied and the dislocation bends, the vertical component from the line tensions, which acts directly against the force caused by the shear stress, grows as the dislocation approaches a semicircular shape. When the dislocation becomes a semicircle, all of the line tension is acting against the bending force induced by the shear stress, because the line tension is perpendicular to the horizontal
Horizontal plane
In geometry, physics, astronomy, geography, and related sciences, a plane is said to be horizontal at a given point if it is perpendicular to the gradient of the gravity field at that point— in other words, if apparent gravity makes a plumb bob hang perpendicular to the plane at that point.In...
between A and B. For the dislocation to reach this point, it is thus evident that the equation:
must be satisfied, and from this we can solve for the shear stress:
This is the stress required to generate dislocation from a Frank-Read source. If the shear stress increases any further and the dislocation passes the semicircular equilibrium state, it will spontaneously continue to bend and grow, spiraling around the A and B pinning points, until the segments spiraling around the A and B pinning points collide and cancel. The process results in a dislocation loop around A and B in the slip plane which expands under continued shear stress, and also in a new dislocation line between A and B which, under renewed or continued shear, can continue to generate dislocation loops in the manner just described. A Frank-Read loop can thus generate many dislocations in a plane in a crystal under applied stress. The Frank-Read source mechanism explains why dislocations are primarily generated on certain slip planes; dislocations are primarily generated in just those planes with Frank-Read sources. It is important to note that if the shear stress does not exceed:
and the dislocation does not bend past the semicircular equilibrium state, it will not form a dislocation loop and instead revert to its original state.