View source for Quantized state systems method

{{Orphan|date=June 2013}}

A '''moving crack''' is a [[Fracture|crack]] that propagates with some speed due to loading and unloading of a metal work material.{{Clarify|date=May 2013}} When loading and unloading is being done, a large fraction of irreversible energy associated with those actions is eventually dissipated as [[heat]] and other part is being stored in the work material due to change in material structure and constraints caused dislocation pile-ups, etc.<ref>{{cite book|last=Ewalds|first=H.L.|coauthors=Wanhill, R.J.H.|title=Fracture Mechanics|publisher=Edward Arnold|year=1984|ISBN=0-7131-3515-8}}</ref> The fraction of heat dissipated in material (<math>\beta</math>) depends on type of material of fracture model is being used there. By conducting some experiments, it is found that the fraction <math>\beta</math> dissipated as heat may be as large as 0.85-0.95 for metals,<ref>{{cite journal|first=G. I.|last=Taylor|coauthors=H. Quinney|year=1934|title=The Latent Energy Remaining in a Metal after Cold Working|journal=[[Proceedings of the Royal Society A]] |volume=143|pages=307–326|doi=10.1098/rspa.1934.0004}}</ref><ref name="Bever">{{cite journal|first=M.B.|last=Bever|coauthors=D.L. Holt and A.L. Titchener|year=1973|title=The stored energy of cold work|journal=Prog. Mat. Sci.|volume=17|issue=1}}</ref> but this value doesn't depend on the magnitude and rate of deformation.<ref name="Bever" /><ref>{{cite journal|last=Mason|first=J.J.|coauthors=Rosakis, A.J., and Ravichandran, G.|year=1994|title=On the strain and strain rate dependence of the fraction of plastic work converted into heat: an experimental study using high speed infrared detectors and the Kolsky bar|journal=Mechanics of Materials|volume=17|pages=135–145}}</ref> So according to work of Mason, <math>\beta</math> can be as small as 0.5 for [[aluminium]] and [[steel]] at low value of strains only and for [[titanium]] at both and low value of strains.

== Heat generation and temperature increment ==

In general, the amount of loading and unloading energy which is converted into heat is for unit volume having high value. So this large value of heat generation per unit of volume results in substantial rise in [[temperature]] of the tip of the moving crack. This temperature rise can be of several hundred degrees [[Celsius]] as found in experiment done by Mason and Rosakis<ref>{{cite journal|first=J. J.|last=Mason|coauthors=A. J. Rosakis|year=1993|title=On the Dependence of the Dynamic Crack Tip Temperature Fields in Metals Upon Crack Tip Velocity and Material Parameters|journal=SM Report|volume=92|issue=3}}</ref> and others. The process region near tip of moving crack is the zone for maximum temperature.<ref name="sciencedirect">{{cite journal|first=F.|last=D’Amico|coauthors=G. Carbone, M.M. Foglia, and U. Galietti|title=Moving cracks in viscoelastic materials: Temperature and energy-release-rate measurements|journal=Engineering Fracture Mechanics|volume=98|year=2013|pages=315–325|ISSN=0013-7944|doi=10.1016/j.engfracmech.2012.10.026}}</ref>

== Temperature measurement ==

Mostly, for measurement of temperature at tip of moving crack mode-I of fracture is being preferred. But maximum temperatures are expected for other two modes-(I, II)of fracture due large deformations with [[shear band]]ing, particularly with high confining [[pressure]], where impact generated shear bands is due to the impact load itself.<ref name="sciencedirect"/> According to a study by Zhou et al.<ref>{{cite journal|first=S.J|last=Zhou|coauthors=P.S. Lomdahl, R. Thomson and B.L. Holian|year=1996|title=Dynamic crack processes via molecular dynamics|journal=Phys. Rev. Lett.|volume=76|pages=2318–2321}}</ref> and Rosakis et al.(1997)<ref>Rosakis et al.(1997)</ref> on impact produced shear bands of mode-II gives a result of temperature rises of over 1650 [[Kelvin|K]] for C-300 steel. These temperatures are being measured by special purpose high precision [[thermocouple]].

== Mathematical formulation of temperature of moving crack tip ==

If one applies the cell model of material and given total energy supply as <math>A</math> to a central cell (this can be found by the area of a cohesion-decohesion curve){{Citation needed|date=May 2013}}, with heat generated = <math>\beta A</math>, where <math>\beta</math> = fraction of energy supplied converted into heat energy and considering adiabatic temperature rise i.e. no heat is going out by [[Thermal conduction|conduction]], temperature rise becomes:

<div style="text-align: center;"><math>T=\frac{\beta A}{\rho c d^3}</math></div>

where <math>\rho</math> = density, <math>c</math> = specific heat, <math>d^3</math> = volume of the cell.<ref>Yunus A. Cengel - Thermodynamics</ref> The above calculation of T conduction of heat from the body has been neglected and this assumption is not valid.

Revising this for conduction using the moving crack tip governing equation for heat conduction for the upper half of the crack (<math>y>0</math>, where <math>y=0</math> is the plane passing through crack) in <math>x</math>-direction is

<div style="text-align: center;"><math>\Delta T - \frac{1}{a^2} \frac{\partial T}{\partial t} = - \frac{1}{\lambda} \frac{\partial Qv}{\partial t}</math><ref>Heat transfer by J.P.Holman (conduction equation)</ref></div>

where  <math>T</math> is temperature at time <math>t</math>, <math>\lambda</math> is conductivity of material, <math>a^2</math> is the diffusivity of material <math>\frac{\lambda}{\rho c}</math>, and 
<math>Qv(x,y,z,t)</math> is the heat per unit volume. Solving this heat conduction equation using [[Laplace transform]], one gets

<div style="text-align: center;"><math>T = \frac{Qv}{\rho c} \operatorname{erf} \left(\frac{h}{4 a \sqrt{t}}\right)</math></div>

where, <math>\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}\int_{0}^z e^{-t^2}\,\mathrm dt.</math> is the [[error function]].

Using the final equation of <math>T</math>, one can calculate the temperature at the tip of the moving crack.

== References ==

{{Reflist}}
<!-- Be sure to cite all of your sources in <ref>...</ref> tags and they will automatically display when you hit save. The more reliable sources added the better! See [[Wikipedia:REFB]] for more information-->

{{improve categories|date=June 2013}}

[[Category:Metalworking]]
[[Category:Metallurgy]]