IdeaWazaWiki - User contributions [en]

About fracture mechanics

2008-04-29T01:05:21Z

18.95.6.179:

{{refimprove||date=June 2006}}

'''Fracture mechanics''' is a method for predicting failure of a structure containing a crack. 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.

In modern [[materials science]], fracture mechanics is an important tool in improving the mechanical performance of materials and components. It applies the [[physics]] of [[stress (physics)|stress]] and [[Strain (materials science)|strain]], in particular the theories of [[Elasticity (physics)|elasticity]] and [[plasticity (physics)|plasticity]], to the microscopic [[crystallographic defect]]s found in real materials in order to predict the macroscopic mechanical failure of bodies. [[Fractography]] is widely used with fracture mechanics to understand the causes of failures and also verify the theoretical failure predictions with real life failures.

==The need for fracture mechanics==
[[Image:Tay1.jpg|thumb|right|Tay Bridge Disaster (1879)]]

In many cases, failure of engineering structures through fracture can be fatal; one example is that of the [[Tay Rail Bridge]] disaster (right). Often disasters occur because engineering structures contain cracks - arising either during production or during service (e.g. from [[Fatigue (material)|fatigue]]). For instance, growth of cracks in pressure vessels due to crack propagation could cause a fatal explosion. If failure were ever to happen, we would rather it were by [[yield (engineering)|yield]] or by leak before break.

Since cracks can lower the strength of the structure beyond that due to loss of load-bearing area a material property, above and beyond conventional strength, is needed to describe the fracture resistance of engineering materials. This is the reason for the need for fracture mechanics - the evaluation of the strength of cracked structures.

==The history of fracture mechanics==
===Griffith's energy relation===
Fracture Mechanics was invented during World War I by English aeronautical engineer, [[Alan Arnold Griffith|A.A.Griffith]], to explain the failure of brittle materials. Griffith was faced with the problem that theoretical calculations showed that the stress at the tip of a sharp crack approaches infinity. Accordingly, any structure containing a crack should fail, no matter how small the crack or how light the load. To solve this dilemma, Griffith developed a thermodynamic approach. He assumed that growth of a crack requires creation of surface energy, which is supplied by the loss of strain energy accompanying the relaxation of local stresses as the crack advances. Failure occurs when the loss of strain energy is sufficient to provide the increase in surface energy.

===Irwin's modification of Griffith's energy relation===
[[Image:TankerSchenectady.jpg|thumb|right|The [[S.S. Schenectady|S.S. ''Schenectady'']] split apart by [[brittle fracture]] while in harbor (1944)]]

Griffith’s work was ignored for over twenty years until a group under [[G.R. Irwin]] at the U.S. Naval Research Laboratory (NRL) took it up during World War II. Irwin and his colleagues developed a modified form of Griffith's approach; they reformulated it in terms of stress, rather than energy. Their work resulted in a new materials property, [[fracture toughness]], which is denoted ''KIc'', and is now universally accepted as the defining property of fracture mechanics.

But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo [[plastic deformation]] at the tip of a crack violating the underlying assumption of the theory. Linear-elastic fracture mechanics is of limited practical use for structural steels for two other reasons:

(1) Fracture toughness testing is very expensive and sufficient information for selection of steels can be obtained from the simpler and cheaper [[Charpy impact test]]

(2) If a part's response to load is sufficiently close to linear-elastic that ''KIc'' can be measured, there is little plastic relaxation at the crack tip and the steel will be [[brittle]]. Structural steels, in particular, can be prone to brittle fracture, which has led to a number of catastrophic failures.

==Elastic-plastic fracture mechanics==
[[Image:Aircraft Crash.jpg|thumb|right|[[Vertical stabilizer]], which separated from the aircraft leading to a fatal crash(2001)]]

In the mid-1960s J.R. Rice (then at Brown University) 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 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 Equation (3.1) of the Appendix to this article. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior.

==Fully plastic fracture mechanics==
If the alloy is so tough that the yielded region ahead of the crack extends to the far edge of the specimen before fracture, the crack is no longer an effective stress concentrator. Instead, the presence of the crack merely serves to reduce the load-bearing area. In this regime the failure stress is conventionally assumed to be the average of the yield and ultimate strengths of the alloy.

==Engineering applications of fracture mechanics==
The following information is needed for a fracture mechanics prediction of failure:
*Applied load
*Residual stress
*Size and shape of the part
*Size, shape, location, and orientation of the crack

Usually not all of this information is available and conservative assumptions have to be made.

Occasionally post-mortem fracture-mechanics analyses are carried out. In the absence of an extreme overload, the causes are either insufficient toughness (KIc) or an excessively large crack that was not detected during routine inspection.

==Short summary==
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 [[structural failure|failure]] of the flawed structure. Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new. There is a high demand for engineers with fracture mechanics expertise - particularly in this day and age where engineering failure is considered 'shocking' amongst the general public.

==Appendix: mathematical relations==
===Griffith's crack theory: strain energy release rate===
For the simple case of a thin rectangular plate with a crack perpendicular to the load Griffith’s theory becomes:

:<math>G = \frac{\pi \sigma^2 a}{E}\,</math>                 (1.1)

where <math>G</math> is the strain energy release rate, <math>\sigma</math> is the applied stress, <math>a</math> is half the crack length, and <math>E</math> is the [[Elastic modulus|Young’s modulus]]. The strain energy release rate can otherwise be understood as: the rate at which energy is absorbed by smoking of the crack rock.

However, we also have that:

:<math>G_c = \frac{\pi \sigma_f^2 a}{E}\,</math>                 (1.2)

If <math>G</math> ≥ <math>G_c</math>, this is the criterion for which the crack will begin to propagate.

===Irwin's modified Griffith crack theory: fracture toughness===
Eventually a modification of Griffith’s solids theory emerged from this work; a term called [[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:

:<math>K_I = \sigma \sqrt{\pi a}\,</math>                 (2.1)

and

:<math>K_c = \sqrt{E G_c}\,</math> (for [[plane stress]])                 (2.2)

:<math>K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\,</math> (for [[plane strain]])                 (2.3)

where ''K''I is the [[stress intensity]], ''Kc'' the [[fracture toughness]], and <math>\nu</math> is [[Poisson ratio|Poisson’s ratio]]. It is important to recognize the fact that fracture parameter ''K''c has different values when measured under plane stress and plane strain

Fracture occurs when <math>K_I \geq K_c</math>. For the special case of plane strain deformation, <math>K_c</math> becomes <math>K_{Ic}</math> 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:

[[Image:Fracture_modes_v2.svg|thumb|The three fracture modes.]]
There are three ways of applying a force to enable a crack to propagate: 
*'''Mode I crack''' – Opening mode (a [[tensile stress]] normal to the plane of the crack)
*'''Mode II crack''' – Sliding mode (a [[shear stress]] acting parallel to the plane of the crack and perpendicular to the crack front)
*'''Mode III crack''' – Tearing mode (a [[shear stress]] acting parallel to the plane of the crack and parallel to the crack front)

We must note that the expression for <math>K_I</math> in equation 2.1 will be different for geometries other than the center cracked plate, as discussed in the article on [[stress intensity]]. Consequently, it is necessary to introduce a [[dimensionless number|dimensionless correction factor]], ''Y'', in order to characterize the geometry. We thus have:

:<math>K_I = Y \sigma \sqrt{\pi a}\,</math>                 (2.4)

where ''Y'' is a function of the crack length and width of sheet given by:

:<math>Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\,</math>                 (2.5)

for a sheet of finite width ''W'' containing a through-thickness crack of length 2''a'', or

:<math>Y \left ( \frac{a}{W} \right ) = 1.12 - \frac{0.41}{\sqrt \pi} \frac{a}{W} + \frac{18.7}{\sqrt \pi} \left ( \frac{a}{W} \right )^2 - \cdots\,</math>                 (2.6)

for a sheet of finite width ''W'' containing a through-thickness edge crack of length ''a''

===Elastic-plastic fracture mechanics theory===
Since engineers became accustomed to using ''K''Ic to characterise fracture toughness, a relation has been used to reduce ''J''Ic to it:

:<math>K_{Ic} = \sqrt{E^* J_{Ic}}\,</math>          where <math>E^* = E</math> for plane strain and <math>E^* = \frac{E}{1 - \nu^2}</math> for plane stress          (3.1)

The remainder of the mathematics employed in this approach is interesting, but is probably better summarised in external pages due to its complex nature (refer to the Useful Websites section).

==References==
*C. P. Buckley, "Material Failure", Lecture Notes (2005), [[University of Oxford]]
*T. L. Anderson, "Fracture Mechanics: Fundamentals and Applications" (1995) CRC Press.

==See also==
*[[fatigue (material)|Fatigue]]
*[[Stress corrosion cracking]]
*[[Stress intensity factor]]

==External links==
*[http://www.efunda.com/formulae/solid_mechanics/fracture_mechanics/fm_intro.cfm eFunda - Fracture Mechanics]
*[http://www2.umist.ac.uk/material/research/intmic/features/charpy/notes.htm UMIST - Charpy Impact Test]
*[http://www.engin.brown.edu/courses/EN175/Notes/Failure_Plasfrac/Failure_Plasfrac.htm Brown University Engineering - Mathematical Relations]
*[http://dspace.library.cornell.edu/handle/1813/3075 Fracture Mechanics Notes from Cornell University]

[[Category:Mechanical engineering]]
[[Category:Structural engineering]]
[[Category:Materials science]]
[[Category:Continuum mechanics]]
[[Category:Fracture mechanics]]

[[de:Bruchmechanik]]
[[ja:破壊力学]]
[[ru:Механика разрушения твёрдых тел]]
[[fi:Murtumismekaniikka]]
[[sv:Brott (mekanik)]]
[[uk:Механіка руйнування]]
[[zh:断裂力学]]

Timeline of material advances

2008-01-06T02:44:45Z

18.95.6.208: New page: == Prehistoric to 0 CE == === ~8000 BCE - Earliest Form of Metallugry === Old Worl Neolithic peoples decorate copper by hammering === ~28000 BCE - First Fired Ceramics === === ~5000 BC...

== Prehistoric to 0 CE ==

=== ~8000 BCE - Earliest Form of Metallugry ===

Old Worl Neolithic peoples decorate copper by hammering

=== ~28000 BCE - First Fired Ceramics ===

=== ~5000 BCE - Discovery of Ability to Extract Liquid Copper from Malachite and Azurite ===

First example of extractive metallurgy

=== ~3500 BCE - Smelting of Iron by Egyptians ===

Iron is the most dominant metallurgical material

=== ~3000 BCE - Combination of Tin Ore and Copper Ore to Produce Bronze ===

Discovered in area of modern Syria and Turkey

=== ~2200 BCE - Invention of Glass in Iran ===

A great nonmetallic engineering material

=== ~1500 BCE - Lost-Wax Casting ===

Developed by Metal Workers in Near East

=== ~1500 BCE Porcelain ===

Crafted by Potters in China

=== ~300 BCE - Development of Crucible Steel Making in India ===

=== ~200 BCE - Iron Casting===

Introduced in China

=== ~100 BCE - Glass Blowing===

Likely Developed by Phoenicians

== 0 CE - 999 CE ==

=== 400 - Seven Meter High Iron Pillar Forged ===

Created in Dehli, India

== 1000 CE - 1499 CE ==

=== 1450 - Lead-Tin-Antimony Alloy to Cast in Copper Alloy Molds ===

Devised by Johannes Gutenberg

Used in printing press

== 1500 CE - 1599 CE ==

=== 1556 - Examination of Mining and Metallurgy Practiced in 16th Century ===

== 1600 CE - 1699 CE ==

=== ~1668 - Optical Microscopy that Magnifies Greater than 200 Times ===

== 1700 CE - 1799 CE ==

=== 1709 - Replacement of Charcoal by Coke in Blast Furnace in Process of Iron Smelting ===

Discovered by Abraham Darby I

=== 1775 - Invention of Modern Concrete ===

Introduced by John Smeaton

== 1800 CE - 1899 CE ==

=== 1805 - Electroplating ===

Invented by Luigi Brugnatelli

=== 1807 - Foundation of Electrometallurgy and Electrochemistry ===

Process of electrolysis developed by Sir Humphry Davy

Developed to separate element metals from salts

=== 1822 - Theory of Stress and Strain ===

Presented by Augustin Cauchy

=== 1827 - Isolation of Elemental Aluminum ===

Accomplished by Friedrich Wohler

=== 1844 - Vulcanization of Rubber ===

Process invented by Charles Goodyear

=== 1856 - Bottom-Blown Acid Process to Melt Low-Carbon Iron ===

Patented by Bessemer

=== 1863 - Light Microscopy to Study Microstucture of Steel ===

Used by Henry Clifton Sorby

=== 1864 - Periodic Table of Elements ===

Introduced by Mendeleev

=== 1867 - Dynamite ===

Patented by Alfred Nobel

=== 1876 - Basis of Understanding Modern Thermodynamics and Physical Chemistry ===

"On the Equilibrium of Heterogeneous Substances" published by J. Willar Gibbs

=== 1886 - Electrolytic Reduction of Alumina into Aluminum ===

Discovered by Charles Martin Hall and Paul Heroult

=== 1890 - Examination of Microstructure of Hard Steel Alloy ===

Adolf Martens finds banded regions of variously oriented microcrystals

=== 1896 - Discovery of Radioactivity ===

Found by Pierre and Marie Curie

=== 1898 - Development of Phase Diagram of Iron and Carbon ===

Creating by William Roberts-Austen

== 1900 CE - 1949 CE ==

=== 1904 - Alloying Composition of Stainless Steel ===

Developed by Leon Guillet

=== 1909 - Bakelite ===

Synthesized by Leo Baekeland

Bakelite is a thermosetting hard plastic

=== 1911 - Superconductivity ===

Discovered by Kammerlingh Omnes

Found when studying pure metals at low temperatures

=== 1912 - Diffraction of X-Rays by Crystals ===

Discovered by Max von Laue

=== 1913 - Publication of the Bohr Model of Atomic Structure ===

Theory that properties determined by number of electrons in orbits around nucleus

=== 1920 - Statement Regarding Structure of Polymers ===

Work Published by Hermann Staudinger

Polymer Consists of Long Chains of Short Repeating Molecular Units

=== 1920 - Fracture Mechanics ===

Publication by A.A. Griffith of "The Phenomenon of Rupture and Flow of Solids"

Problem of fracture in context of energy balance

=== 1925 - Basis of Quantum Mechanics ===

Matrix mechanics developed by Werner Heisenberg

Wave mechanics and the non-relativistic Schrodinger equation of atoms invented by Erwin Schrodinger

=== 1926 - "Superalloy" ===

Process Patented by Paul Merica

Created by adding small amounts of aluminum to Ni-Cr alloy

=== 1927 - Light-Emitting Diode ===

Paper published by Oleg Losev

=== 1933 - Transmission Electron Microscope ===

Built by Max Knoll and Ernst Ruska

=== 1934 - Theory of Dislocations to Explain Plastic Deformation of Ductile Materials ===

Proposed by Egon Orowan, Michael Polyani, and G.I. Taylor

=== 1935 - Polymer Nylon ===

Wallace Hume Carothers, Julian Hill are among researches who patented the process

=== 1939 - Split Nucleus of Uranium Atom ===

Otto Hahn and Fritz Strassman bombard nucleus of uranium atom by bombarding with neutrons

=== 1939 - Discovery of P- and N-Type Regions in Silicon ===

Found by Russel Ohl, George Southworth, Jack Scaff, and Henry Theuerer

=== 1946 - Metamaterials ===

The term "metamaterials" was coined in 1999 by Rodger M. Walser of the University of Texas at Austin, and he defined metamaterials as "macroscopic composites having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation."

The first metamaterial was developed by W.E. Kock in 1946. He created the metal-lens antennas.

=== 1948 - Transistor ===

Fundamental component of all modern electronics

Invented by John Bardeen, Walter H. Brattain, and William Schockley

== 1950 CE - 1959 CE ==

=== 1958 - Carbon Fibers ===

Dr. Roger Bacon created the first high performance carbon fibers at the Parma Technical Center outside of Cleveland, OH

=== 1958 - "Microchip" ===

Created by Jack Kilby

Integration of Capacitors, Resistors, Diodes, and Transistors in Germanium

=== 1959 - Introduction of the concepts of nanotechnology ===

Presentation by Richard Feynman of "There's Plenty of Room at the Bottom"

=== 1959 - Photolithography ===

== 1960 CE - 1969 CE ==

=== 1962 - Laser Diode ===

The first to demonstrate coherent light emission from a semiconductor diode (the first laser diode), is widely acknowledged to have been Robert N. Hall and his team at the General Electric research center in 1962.

=== 1965 - Commercial Scanning Electron Microscope ===

Introduced by Cambridge Instruments

== 1970 CE - 1979 CE ==

=== 1976 - Lithium Ion Batteries ===

Paper published by M.S. Whittingham: "Electrical Energy Storage and Intercalation Chemistry"

== 1980 CE - 1989 CE ==

=== 1981 - Scanning Probe Microscopes ===

Scanning probe microscopy (SPM) is a new branch of microscopy that forms images of surfaces using a physical probe that scans the specimen.

=== 1988 - Giant Magnetoresistive Effect ===

GMR was independently discovered in Fe/Cr/Fe trilayers by a research team led by Peter Grünberg of the Jülich Research Centre (DE), who owns the patent, and in Fe/Cr multilayers by the group of Albert Fert of the University of Paris-Sud (FR), who first saw the large effect in multilayers that led to its naming, and first correctly explained the underlying physics.

== 1990 CE - 1999 CE ==

=== 1991 - Discovery of Nanotubes ===

Found by Sumio Iijima

=== 1994 - The International Technology Roadmap of Semiconductors ===

== 2000 CE - Present ==

=== 2001 National Nanotechnology Initiative ===

A United States federal initiative of nanoscale science and technology

== Source ==

"[http://www.sciencedaily.com/releases/2007/12/071218101208.htm Top 10 Advances In Materials Science Over Last 50 Years]" Science Daily. December 19, 2007

"[http://www.materialmoments.org/vote.html The Greatest Moments in Materials Science and Engineering]" JOM